A binary string is a non-empty sequence of $$$0$$$'s and $$$1$$$'s, e.g., 010110, 1, 11101, etc. The edit distance of two binary strings $$$S$$$ and $$$T$$$, denoted by $$$edit(S, T)$$$, is the minimum number of single-character edit (insert, delete, or substitute) to modify $$$S$$$ into $$$T$$$. For example, the edit distance of 0011 and 1100 is $$$4$$$, i.e. 0011 $$$\rightarrow$$$ 011 $$$\rightarrow$$$ 11 $$$\rightarrow$$$ 110 $$$\rightarrow$$$ 1100. The edit distance of 1100101 and 1110100 is $$$2$$$, i.e. 1100101 $$$\rightarrow$$$ 1110101 $$$\rightarrow$$$ 1110100.

Ayu has a binary string $$$S$$$. She wants to find a binary string with the same length as $$$S$$$ that maximizes the edit distance with $$$S$$$. Formally, she wants to find a binary string $$$T_{max}$$$ such that $$$|S| = |T_{max}|$$$ and $$$edit(S, T_{max}) \geq edit(S, T')$$$ for all binary string $$$T'$$$ satisfying $$$|S| = |T'|$$$.

She needs your help! However, since she wants to make your task easier, you are allowed to return any binary string $$$T$$$ with the same length as $$$S$$$ such that the edit distance of $$$S$$$ and $$$T$$$ is more than half the length of $$$S$$$. Formally, you must return a binary string $$$T$$$ such that $$$|S| = |T|$$$ and $$$edit(S, T) \gt \frac{|S|}{2}$$$.

Of course, you can still return $$$T_{max}$$$ if you want, since it can be proven that $$$edit(S, T_{max}) \gt \frac{|S|}{2}$$$ for any binary string $$$S$$$. This also proves that there exists a solution for any binary string $$$S$$$. If there is more than one valid solution, you can output any of them.

Andi has $$$N$$$ gears (numbered from 1 to $$$N$$$) on a board. Each gear has an arrow initially pointing upward. Some pair of gears may touch each other. If we construct a graph where each gear is a node, and we connect each pair of touching gears with an edge, then the structure of this graph is a tree. For example, the following is a possible gear configuration example with $$$N = 4$$$ gears and gear $$$2$$$ touches all other gears.

Standard gear rotation rule applies: Suppose a gear $$$u$$$ touches another gear $$$v$$$, and gear $$$u$$$ is rotated $$$\alpha$$$ degrees clockwise, then gear $$$v$$$ will be rotated $$$\alpha$$$ degrees counter-clockwise, and vice-versa.

Andi wants to perform three kinds of operations:

1. Take out a gear from its position on the board.
2. Place a gear back to its original position on the board. When doing this, Andi places a gear back in a way such that the arrow points to the same degree as when it was removed. Assume that it is always possible to do that, i.e. you do not need to concern with the mechanical aspect of the gear.
3. Rotate a gear $$$\alpha$$$ degrees clockwise.

Let $$$\delta_u$$$ be the arrow's degree (clockwise, modulo $$$360$$$) of gear $$$u$$$ after $$$Q$$$ operations are done. Andi wants to know the sum of $$$\delta_u$$$ for all $$$u$$$. Furthermore, since rotating gears requires a lot of work, Andi also wants to know how much energy he needs for every Type 3 operation (rotating a gear). The amount of energy Andi needs to perform the Type 3 operation is defined as $$$\langle$$$number of rotating gears$$$\rangle$$$ $$$\times$$$ $$$\langle$$$rotation done in degrees$$$\rangle$$$

Ayu managed to steal Budi's treasure box and ready to uncover any secret Budi hides. Unfortunately, the treasure box has some security system to prevent any unauthorized person (e.g., Ayu) from opening it.

To unlock the treasure box, Ayu has to input a correct PIN (Personal Identification Number) of length $$$N$$$, which of course, she doesn't have. Ayu has no choice other than trying all possible PIN combinations. However, Ayu notices that this security system has an interesting (old) mechanism.

When you enter an $$$N$$$ digits PIN, it is evaluated automatically and promptly, i.e. you don't need to push some "enter" button to confirm the PIN. Whenever your entered PIN is wrong, it removes the oldest (first) digit and shifts all the remaining to the left, thus, you only need to enter one more (last) digit to make it $$$N$$$ length again.

For example, let $$$N = 4$$$. If we input 204320435, then we actually test 6 PINs (with 5 different PINs):

• , tested PIN = 2043
• , tested PIN = 0432
• , tested PIN = 4320
• , tested PIN = 3204
• , tested PIN = 2043
• , tested PIN = 0435

As a CS student, Ayu knows that finding the correct PIN by trying all possible combinations can be very time-consuming, but alas, there's no other way. Ayu decides that she wants to test at least $$$K$$$ different PINs on the first day. Your task is to help Ayu by simply giving her the string $$$S$$$ which contains at least $$$K$$$ different PINs. Ayu doesn't care which PIN she's going to test (so long at least there are $$$K$$$ different PINs) or whether any PIN is tested more than once in $$$S$$$, but string $$$S$$$ needs to be as short as possible. If there is more than one possible string for $$$S$$$, you can output any of them.

Note that, as this system is quite old, there are only $$$M$$$ available digits ranging from $$$0$$$ to $$$9$$$.

You are given a board of $$$R$$$ rows and $$$C$$$ columns where each cell is either a dry land ('#') or an icy land ('.'). You can move to either one of the four directions (north, south, east, or west) in the board.

You can walk or standstill on any dry land without any problem; however, an icy land is very slippery. If you stand at cell $$$(r,c)$$$ and decide to move on a particular direction, then you will move on that direction continuously until you reach a dry land or the border of the board.

For example, consider the following board of $$$6$$$ rows and $$$7$$$ column.

   #......
   .#.....
   ..##...
   ..##...
   .......
   .#.....

• From $$$(1,3)$$$, the $$$1^{st}$$$ row and $$$3^{rd}$$$ column, if you move to the south, then you will end up at $$$(3,3)$$$.
• From $$$(4,4)$$$, if you move to the north, then you will end up at $$$(3,4)$$$.
• From $$$(1,1)$$$, if you move to the east, then you will end up at $$$(1,7)$$$.
• From $$$(6,5)$$$, if you move to the west, then you will end up at $$$(6,2)$$$.

Supposed your initial position is at cell $$$(3,7)$$$ and your moves are west, west, west, south, east, north, west, and west, respectively. Then your positions are $$$(3,7)$$$ $$$\rightarrow$$$ $$$(3,4)$$$ $$$\rightarrow$$$ $$$(3,3)$$$ $$$\rightarrow$$$ $$$(3,1)$$$ $$$\rightarrow$$$ $$$(6,1)$$$ $$$\rightarrow$$$ $$$(6,2)$$$ $$$\rightarrow$$$ $$$(2,2)$$$ $$$\rightarrow$$$ $$$(2,1)$$$ $$$\rightarrow$$$ $$$(2,1)$$$, thus, visiting $$$8$$$ different cells. Note that cell $$$(2,1)$$$ is visited twice. Cells in which you only passed through are also not considered as visited, e.g., in the example above, if you move south from $$$(3,1)$$$, then you will pass through $$$(4,1)$$$ and $$$(5,1)$$$, and arrive at $$$(6,1)$$$; thus, $$$(4,1)$$$ and $$$(5,1)$$$ are not considered as visited from that particular move, only $$$(3,1)$$$ and $$$(6,1)$$$ are.

You are allowed to change any icy land into a dry land, and your goal is to make sure that you can always visit all the cells in the board regardless of your initial starting position; in other words, you do not know your starting position yet, but given the board, you want to make sure that you can achieve your goal. What is the minimum number of cells you need to change to ensure that your goal can be achieved?

Sauville has been engulfed in a devastating war with its neighboring country, Norland, for centuries. Their border can be represented as a straight horizontal line $$$y = 0$$$, where all areas for $$$y \lt 0$$$ belong to Sauville and all areas for $$$y \gt 0$$$ belong to Norland.

Norland has deployed $$$N$$$ of its artilleries at position $$$(x_i, y_i)$$$ where $$$y_i \gt 0$$$. In addition, Norland also has built $$$M$$$ defensive walls. Each defensive wall can be represented as a tuple $$$\langle x_{j1}, x_{j2}, y_j \rangle$$$, which is a horizontal line segment spanned from $$$(x_{j1}, y_j)$$$ to $$$(x_{j2}, y_j)$$$, where $$$x_{j1} \lt x_{j2}$$$ and $$$y_j \gt 0$$$. It is known to Sauville that no Norland's artillery is located at any of its defensive walls (including its endpoints), and no two artilleries are at the same position. It is also known that no two walls (including their endpoints) are sharing a common point.

Sauville has decided to build a watchtower to observe any suspicious activity on any Norland's artilleries. As the cost to build one watchtower is almost astronomical for Sauville, they can only afford to build one. Thus, $$$Q$$$ position candidates $$$(x_k, y_k)$$$ where $$$y_k \lt 0$$$ for the watchtower have been proposed. If the watchtower is built at $$$(x, y)$$$, then all artilleries which lie on the line-of-sight from $$$(x, y)$$$ can be observed (visible) by the watchtower. A position $$$(x_i, y_i)$$$ lies on the line-of-sight from $$$(x, y)$$$ if and only if the straight line connecting $$$(x_i, y_i)$$$ and $$$(x, y)$$$ does not intersect with any defensive walls (including its endpoints); in other words, there should be no point $$$(x', y')$$$ such that $$$(x', y')$$$ lies on a defensive wall and also on the line segment between $$$(x_i, y_i)$$$ and $$$(x, y)$$$. Note that an artillery does not affect the visibility of any other artilleries from the watchtower.

Your task in this problem is to determine the number of Norland's artilleries which can be observed from each proposed watchtower position.

Ayu is participating in ABC 2018 (Arranging Blocks Competition). In this competition, each contestant is given $$$M$$$ minutes and $$$N$$$ tasks which should be solved one-by-one in the given order. The contestant who solves the most tasks is the winner. What makes this contest interesting to you (ICPC contestants) is that this contest uses a similar encouragement as ICPC, i.e. balloons. In particular, each time a contestant solves a task, s/he will be given a balloon.

Ayu is convinced that she can defeat all other contestants, except one particular contestant, Budi, her rival. Ayu knows well her skill, i.e. she knows exactly how long it takes for her to solve a particular task. Unsurprisingly, Ayu also knows well Budi's skill (they are rival for a reason). Let there be two arrays of integers $$$A_{1..N}$$$ and $$$B_{1..N}$$$. $$$A_i$$$ denotes the time (in minutes) needed by Ayu to solve the $$$i^{th}$$$ task, while $$$B_i$$$ denotes the time (in minutes) needed by Budi to solve the same $$$i^{th}$$$ task.

Here comes the interesting part. Ayu knows that Budi is very sensitive to any disturbingly loud sound like a balloon being popped. Whenever Budi is surprised (due to a balloon being popped), he will lose his concentration and has to repeat the task he's doing. For example, suppose Budi needs $$$10$$$ minutes to solve a particular task. If a balloon pops at the $$$7^{th}$$$ minute, then Budi repeats the task at the $$$7^{th}$$$ minute (out of his $$$10$$$ minutes), causing him to solve the task with $$$7 + 10 = 17$$$ minutes. If two balloon pops, each at the $$$7^{th}$$$ and $$$13^{th}$$$ minute, then Budi repeats the task at minute $$$7^{th}$$$ (out of his $$$10$$$ minutes), repeats it again at minute $$$6^{th}$$$ (out of his $$$10$$$ minutes), and finally solved the task with $$$7 + 6 + 10 = 23$$$ minutes. If a balloon pops at the same time Budi supposed to solve the task (i.e. at the $$$10^{th}$$$ minute in this example), then Budi will also not solve that task. Therefore, Budi has to spend another $$$10$$$ minutes (for a total of $$$10 + 10 = 20$$$ minutes) to solve that particular task in this case.

Ayu plans to exploit Budi's weakness in order to defeat him, i.e. Ayu will strategically use the balloons (popping them at integer minutes) she gets from solving the tasks. Your task in this problem is to find out whether it is possible for Ayu to have the number of solved tasks to be strictly larger than Budi's. If it is possible, you should output one (any) working plan on when she should pop the balloons.

Note that if Ayu solves a task at exactly the $$$M^{th}$$$ minute, then the task is considered as solved. Similarly, if Budi solves a task at exactly the $$$M^{th}$$$ minute, then the task is considered as solved, except if Ayu decides to pop a balloon at the same time. Also, Ayu can pop a balloon as soon as she receives it. Ayu cannot pop more than one balloon at the same minute. She also cannot pop any balloon after the $$$M^{th}$$$ minute mark.

Andi has a network $$$G$$$ with $$$N$$$ machines and $$$M$$$ links; each link connects exactly two different machines. Due to some circumstances, Andi has to make his network "complete", i.e. every machine is directly linked to every other machine. This means Andy has to add a new link between any pair of machines which are not already directly linked.

In order to accomplish his goal, Andi outsources the work to a company, Go. Go accepts any work order on a network $$$G$$$ with a requested integer $$$k$$$, i.e. $$$go(G,k)$$$. The way Go works: First, it creates a list $$$L$$$ containing all pair of machines which are not yet directly linked. Then, Go evaluates each pair of machines $$$(a,b)$$$ from $$$L$$$ and create a new link between them if $$$\delta_a + \delta_b \ge k$$$. Note that $$$\delta_a$$$ is the degree of machine $$$a$$$, i.e. the number of links $$$a$$$ has at the time of evaluation. Similarly, $$$\delta_b$$$ is for machine $$$b$$$.

The problem with Go's procedure is it evaluates each pair of machines in the order appeared in $$$L$$$. For example, let $$$G$$$ be a network with $$$N = 4$$$ machines (machine $$$1$$$, $$$2$$$, $$$3$$$, and $$$4$$$) and $$$M = 3$$$ links; the links are $$$\{(1, 2), (2, 3), (3, 4)\}$$$. The degree of each machine before a work order is requested is: $$$\delta_1 = 1$$$, $$$\delta_2 = 2$$$, $$$\delta_3 = 2$$$, $$$\delta_4 = 1$$$, or simply can be written as $$$[1, 2, 2, 1]$$$. Let say a work order with $$$k = 3$$$ is requested ($$$go(G,3)$$$).

Consider these two lists:

• $$$L_1 = ((1,4),(1,3),(2,4))$$$.
  • Evaluate $$$(1,4)$$$ and don't create a new link since $$$\delta_1 + \delta_4 = 1 + 1 = 2$$$. The degree is still $$$[1, 2, 2, 1]$$$.
  • Evaluate $$$(1,3)$$$ and create a new link since $$$\delta_1 + \delta_3 = 1 + 2 = 3$$$. The degree becomes $$$[2, 2, 3, 1]$$$.
  • Evaluate $$$(2,4)$$$ and create a new link since $$$\delta_2 + \delta_4 = 2 + 1 = 3$$$. The degree becomes $$$[2, 3, 3, 2]$$$.
  The final result is a network with $$$5$$$ links, which is not complete as it misses the $$$(1,4)$$$ link.
• $$$L_2 = ((2,4),(1,3),(1,4))$$$.
  • Evaluate $$$(2,4)$$$ and create a new link since $$$\delta_2 + \delta_4 = 2 + 1 = 3$$$. The degree becomes $$$[1, 3, 2, 2]$$$.
  • Evaluate $$$(1,3)$$$ and create a new link since $$$\delta_1 + \delta_3 = 1 + 2 = 3$$$. The degree becomes $$$[2, 3, 3, 2]$$$.
  • Evaluate $$$(1,4)$$$ and create a new link since $$$\delta_1 + \delta_4 = 2 + 2 = 4$$$. The degree becomes $$$[3, 3, 3, 3]$$$.
  The final result is a network with $$$6$$$ links and complete.

Ordering a work to Go can be very expensive with lower $$$k$$$, thus, Andi has to make $$$k$$$ as high as possible while maintaining the possibility that a complete network can be achieved with $$$k$$$. In other words, Andi wants the highest $$$k$$$ such that there exists $$$L$$$ such that $$$go(G,k)$$$ can produce a complete network, and for any $$$j$$$ ($$$j \gt k$$$), there is no valid $$$L$$$ for $$$go(G,j)$$$ which can produce a complete network. Your task in this problem is to find such $$$k$$$.

Andi likes numbers and sequences, especially, sign sequences. A sign sequence is a sequence which consists of $$$-1$$$ and $$$1$$$. Andi is a curious person, thus, he wants to build a sign sequence which length is $$$N$$$ (the positions are numbered from $$$1$$$ to $$$N$$$, inclusive).

However, Andi also likes some challenges. Therefore, he prefilled some positions in the sequence with $$$-1$$$ or $$$1$$$ (the number in these positions cannot be changed). Andi also wants the sequence to fulfill $$$K$$$ constraints. For each constraint, there are 3 numbers: $$$A_i$$$, $$$B_i$$$, and $$$C_i$$$. This means that the sum of numbers which position is in the range $$$[A_i, B_i]$$$ (inclusive) must be at least $$$C_i$$$.

Sounds confusing, right? It is not done yet. Since there can be many sequences that fulfill all the criteria above, Andi wants the sequence to be lexicographically smallest. Sequence $$$X$$$ is lexicographically smaller than sequence $$$Y$$$ if and only if there exists a position $$$i$$$ where $$$X_i \lt Y_i$$$ and $$$X_j = Y_j$$$ for all $$$j \lt i$$$.

Find the sequence Andi wants.

Andi is a young and prominent detective in the police force. His ability to track down criminals, uncover the truth, and solve cases never ceases to amaze all of his colleagues. One day, he is faced with a suspicious eyewitness testimony when working on a certain case. In usual cases, Andi simply ignores such unreliable testimony; however, in this case, the eyewitness testimony is too important to be ignored. To resolve this situation, Andi has to rely on technology, i.e. using a lie detector.

Andi proceeds to use a lie detector to detect whether the eyewitness testimony is true. However, Andi notices that the lie detector he used might have been tampered, thus, he employs a second lie detector to detect whether the first lie detector's result is correct. This situation happens repeatedly such that Andi ends up employing $$$N$$$ lie detectors in total. The $$$i^{th}$$$ lie detector reports the truth of the $$$(i-1)^{th}$$$ lie detector for $$$i = 2..N$$$, and the $$$1^{st}$$$ lie detector reports the truth of the eyewitness testimony.

In the end, Andi knows that the last ($$$N^{th}$$$) lie detector has not been tampered and always report the truth correctly. Now, he needs to determine whether the eyewitness testimony is true given the result of all lie detectors.

For example, let $$$N = 4$$$ and the lie detectors result are $$$(\texttt{LIE},\texttt{LIE},\texttt{TRUTH},\texttt{TRUTH})$$$.

• The $$$4^{th}$$$ lie detector reports that the $$$3^{rd}$$$ lie detector is TRUTH. As the $$$4^{th}$$$ lie detector always report the truth correctly, then the $$$3^{rd}$$$ lie detector's result is correct as it is.
• The $$$3^{rd}$$$ lie detector reports that the $$$2^{nd}$$$ lie detector is TRUTH. As the $$$3^{rd}$$$ lie detector's result is correct as it is, then the $$$2^{nd}$$$ lie detector's result is also correct as it is.
• The $$$2^{nd}$$$ lie detector reports that the $$$1^{st}$$$ lie detector is LIE. As the $$$2^{nd}$$$ lie detector's result is correct as it is, then the $$$1^{st}$$$ lie detector's result is wrong.
• The $$$1^{st}$$$ lie detector reports that the eyewitness testimony is LIE. As the $$$1^{st}$$$ lie detector's result is wrong, then the eyewitness testimony is correct; in other words, what the eyewitness says is true.

Therefore, the eyewitness testimony in this example is true.

Andi is getting married! He and his partner plan to have $$$N$$$ children. To avoid any hassle in the future, Andi wants to decide all their children's name in advance. Specifically, he wants each child to have a name which is lexicographically larger than any of his/her older siblings. Of course, his partner also agrees with this idea. String $$$A$$$ is lexicographically larger than string $$$B$$$ if and only if $$$B$$$ is a prefix of $$$A$$$ or there exists an index $$$i$$$ where $$$A_i \gt B_i$$$ and $$$A_j = B_j$$$ for all $$$j \lt i$$$. Note that a proper name can be as short as one character, but it cannot be an empty string.

Life is good for Andi, until one day he told his soon-to-be-grandmother-in-law (i.e. his partner's grandma) about this marriage plan. After learning that Andi plans to have $$$N$$$ children with her granddaughter, she gave him $$$N$$$ names to be used. Moreover, the $$$i^{th}$$$ name can only be used for the $$$i^{th}$$$ child.

After going through a long debate with her grandma, Andi came into an agreement: The $$$i^{th}$$$ child's name should be a subsequence of the $$$i^{th}$$$ name her grandma gave. A string $$$A$$$ is a subsequence of string $$$B$$$ if $$$A$$$ can be obtained by deleting zero or more characters from $$$B$$$ without changing the remaining characters' order, e.g., ABC is a subsequence of DAEBCCB, but EFG is not a subsequence of FABEGC.

Even though Andi dislikes the given list of names, he still wants to impress his partner by showing that he can satisfy both her grandma's wish and his own desire (i.e. each child's name is lexicographically larger than any of his/her older siblings). However, Andi wonders, what is the maximum possible total length of their children names?

For example, let $$$N = 3$$$, and the names given by his partner's grandma are $$$($$$KARIM$$$,$$$ PARBUDI$$$,$$$ CHANDRA$$$)$$$. Here are several example set of names which satisfies Andi's desire:

• $$$[\texttt{AR}, \texttt{BI}, \texttt{CRA}]$$$ with a total length of $$$2 + 2 + 3 = 7$$$.
• $$$[\texttt{ARI}, \texttt{BUDI}, \texttt{CHANDRA}]$$$ with a total length of $$$3 + 4 + 7 = 14$$$.
• $$$[\texttt{ARIM}, \texttt{ARUDI}, \texttt{CHANDRA}]$$$ with a total length of $$$4 + 5 + 7 = 16$$$.
• $$$[\texttt{AIM}, \texttt{ARBUDI}, \texttt{CHANDRA}]$$$ with a total length of $$$3 + 6 + 7 = 16$$$.
• $$$\cdots$$$

Let $$$G=(V,E)$$$ be a simple undirected graph with $$$N$$$ vertices and $$$M$$$ edges, where $$$V = \{1, \dots, N\}$$$. A tuple $$$\langle u,v,w \rangle$$$ is called as boomerang in $$$G$$$ if and only if $$$\{(u,v), (v,w)\} \subseteq E$$$ and $$$u \ne w$$$; in other words, a boomerang consists of two edges which share a common vertex.

Given $$$G$$$, your task is to find as many disjoint boomerangs as possible in $$$G$$$. A set $$$S$$$ contains disjoint boomerangs if and only if each edge in $$$G$$$ only appears at most once (in one boomerang) in $$$S$$$. You may output any valid disjoint boomerangs, but the number of disjoint boomerangs should be maximum.

For example, consider a graph $$$G = (V,E)$$$ of $$$N = 5$$$ vertices and $$$M = 7$$$ edges where $$$E = \{(1,2)$$$, $$$(1,4)$$$, $$$(2,3)$$$, $$$(2,4)$$$, $$$(2,5)$$$, $$$(3,4)$$$, $$$(3,5)\}$$$.

The maximum number of disjoint boomerangs in this example graph is $$$3$$$. One example set containing the $$$3$$$ disjoint boomerangs is $$$\{\langle 4,1,2 \rangle, \langle 4,3,2 \rangle, \langle 2,5,3 \rangle\}$$$; no set can contain more than $$$3$$$ disjoint boomerangs in this example.

A binary string is a non-empty sequence of $$$0$$$'s and $$$1$$$'s, e.g., 010110, 1, 11101, etc. Ayu has a favorite binary string $$$S$$$ which contains no leading zeroes. She wants to convert $$$S$$$ into its decimal representation with her calculator.

Unfortunately, her calculator cannot work on any integer larger than $$$K$$$ and it will crash. Therefore, Ayu may need to remove zero or more bits from $$$S$$$ while maintaining the order of the remaining bits such that its decimal representation is no larger than $$$K$$$. The resulting binary string also must not contain any leading zeroes.

Your task is to help Ayu to determine the minimum number of bits to be removed from $$$S$$$ to satisfy Ayu's need.

For example, let $$$S$$$ = 1100101 and $$$K = 13$$$. Note that 1100101 is $$$101$$$ in decimal representation, thus, we need to remove several bits from $$$S$$$ to make it no larger than $$$K$$$. We can remove the $$$3^{rd}$$$, $$$5^{th}$$$, and $$$6^{th}$$$ most significant bits, i.e. 1100101 $$$\rightarrow$$$ 1101. The decimal representation of 1101 is $$$13$$$, which is no larger than $$$K = 13$$$. In this example, we removed $$$3$$$ bits, and this is the minimum possible (If we remove only $$$2$$$ bits, then we will have a binary string of length $$$5$$$ bits; notice that any binary string of length $$$5$$$ bits has a value of at least $$$16$$$ in decimal representation).