Given an array of $$$n$$$ integers, your task is to process $$$q$$$ queries of the following types:

• Update: For a given interval $$$[l, r]$$$, update each element $$$a_i$$$ in the interval to $$$(a_i \times c + d)$$$, where $$$c$$$ and $$$d$$$ are given integers.
• Query: Given integers $$$l, r, x$$$, report the number of elements in the interval $$$[l, r]$$$ whose value modulo $$$10^5 + 3$$$ is equal to $$$x$$$, i.e., $$$$$$ \begin{aligned} \#\{i \mid l \le i \le r, \ a_i \bmod (10^5 + 3) = x\} \end{aligned} $$$$$$

An array of $$$n$$$ integers and $$$q$$$ queries are given. Each query consists of two integers $$$(l, r)$$$:

1. Reverse the subarray from index $$$l$$$ to $$$r$$$.
2. Output the number of inversions in the entire array.
3. Restore the array to its original state (i.e., the array as it was before this query)

You are given an array $$$A$$$ of length $$$n$$$ and two integers $$$M$$$ and $$$K$$$. Determine if there exists a sequence $$$P$$$ that is permutation of $$$A$$$ such that for every consecutive pair, $$$$$$ \begin{aligned} (p_i + p_{i+1}^2) \bmod M = K \notag \end{aligned} $$$$$$ If a valid permutation exists, output any one; otherwise, output $$$-1$$$.

Consider the following recursive procedure, written in pseudocode, which transforms an input array into a permutation of its elements:

function generate(sequence b):
    if size of b is 1:
        return b
    else:
        split b into two non-empty contiguous subarrays, b_1 and b_2
        optionally, swap b_1 and b_2
        b_1 = generate(b_1)
        b_2 = generate(b_2)
        concatenate b_1 and b_2 into b', in this order
        return b'
    end if

In other words, at each step, the array is split into two contiguous subarrays (not necessarily of equal length). You may optionally swap the order of the two subarrays before recursing on each. After both subarrays are recursively processed using the same procedure, they are concatenated in their current order to form the result.

Note that the final output of the procedure may vary depending on how the array is split at each step and whether or not a swap is applied.

You are given an array $$$a$$$ of $$$n$$$ positive integers $$$a_1, a_2, \ldots, a_n$$$, which is a permutation of integers from $$$1$$$ to $$$n$$$. Your task is to determine if it is possible for array $$$a$$$ to be the output of generate function when applied to the identity permutation $$$[1,2,\ldots, n]$$$.

In the enchanted kingdom of Arithmora, wizards cast spells using arithmetic expressions. Each spell is a string consisting only of:

• single-digit numbers ("0"–"9"),
• the addition operator ("+"), and
• the multiplication operator ("*").

When a wizard chants the spell, the kingdom's ancient laws dictate that:

1. Multiplication ("*") has higher precedence than addition ("+"), and
2. Both operators are left-associative (evaluated from left to right if they have equal precedence).

For example:

• The spell 2+3*4 is evaluated as $$$2+(3*4)=14$$$.
• The spell 9*1+5*2 is evaluated as $$$(9*1)+(5*2)=19$$$.

Your task is to compute the final value of the given spell expression.

You are given a tree with $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$. Consider the following pseudocode, which performs a preorder traversal of the tree:

a = []
function preorder_traversal(u):
    append u to the back of a
    for each v that is adjacent to u and not visited:
        preorder_traversal(v)

Note that the resulting array $$$a$$$, representing the preorder traversal of the tree starting from node $$$u$$$, may vary depending on the order in which adjacent vertices are visited.

Let $$$S_u$$$ denote the set of all possible arrays $$$a$$$ that can be obtained by running the above traversal starting from vertex $$$u$$$, allowing all possible orderings of adjacent nodes during traversal.

For any two vertices $$$u$$$ and $$$v$$$, define the function $$$f(u, v)$$$ as follows: $$$$$$\min_{a_1 \in S_u, a_2 \in S_v} \text{LCS}(a_1, a_2)$$$$$$ where $$$\text{LCS}(a_1, a_2)$$$ denotes the length of the longest common subsequence between the two arrays $$$a_1$$$ and $$$a_2$$$.

For example, consider a tree with $$$5$$$ vertices and edges $$$(1,2),(1,3), (3,4), (3,5)$$$. Then we have:

• $$$S_1 = \{[1,2,3,4,5],[1,2,3,5,4],[1,3,4,5,2],[1,3,5,4,2]\}$$$
• $$$S_4 = \{[4,3,5,1,2], [4,3,1,2,5]\}$$$

Then we have $$$f(1,4) = 2$$$, because the smallest possible LCS length among all combinations of arrays from $$$S_1$$$ and $$$S_4$$$ is 2.

You are given $$$q$$$ queries. Each query consists of two (not necessarily distinct) vertices $$$x_i$$$ and $$$y_i$$$. For each query, compute the value of $$$f(x_i, y_i)$$$.

In a mechanical power transmission system, several gears are connected by shafts. Each gear transmits torque to the next one. However, due to frictional losses and imperfect contact, only a certain percentage of torque is actually transferred from one gear to another.

The system can be modeled as a directed graph:

• Each gear is a node.
• Each shaft is a directed edge with an efficiency percentage (between 0 and 100).

The torque delivered to the next gear is: $$$$$$ T_{next} = T_{current} \times \frac{\text{efficiency}}{100}. $$$$$$

Your job is to compute the maximum possible torque that can be delivered to a specified target gear, starting from the input gear.

At the university hospital, doctors often prescribe medication to patients based on their weight. To avoid mistakes, prescriptions must be checked against standard safe dosage ranges.

For example, the safe dosage of a certain drug is 10 to 20 milligrams per kilogram of body weight. If a doctor prescribes too little, the treatment may be ineffective; if too much, the patient may suffer side effects.

You are asked to help write a small program that checks whether prescriptions are safe. You are given the weight of a patient (in kilograms) and the prescribed dosage (in milligrams). The safe dosage range for this drug is between 10 and 20 milligrams per kilogram of body weight, inclusive.

Determine whether the prescribed dosage is:

• "Too Low" if it is less than the safe range,
• "Too High" if it is more than the safe range,
• "Safe" if it is within the safe range.