As children, David and Henry used to play "who can name the biggest number?"

Henry: "Twenty!"

David: "Nine thousand!"

Henry: "A million trillion!"

David: "A googolplex squared!"

Henry: "Infinity!"

Determined to win, David takes the game to the next level: "Infinity plus one!"

Lucca, passing by, objects: "How can you count to infinity, let alone past it?"

To give his winning move a mathematical basis, David invokes an ordered set $$$\mathcal S$$$. Only two properties of $$$\mathcal S$$$ are relevant:

• $$$\mathcal S$$$ is well-ordered: every non-empty subset of $$$\mathcal S$$$ has a unique smallest element.
• $$$\mathcal S$$$ is uncountably big: even if we take and name infinitely many elements from $$$\mathcal S$$$, there will always be more.

By taking one smallest element at a time, David identifies numbers with elements of $$$\mathcal S$$$:

• $$$0 = \min(\mathcal S)$$$, the unique smallest element of $$$\mathcal S$$$,
• $$$1 = \min(\mathcal S \setminus \{0\})$$$, the smallest element in the subset that remains when $$$0$$$ is removed from $$$\mathcal S$$$,
• $$$2 = \min(\mathcal S \setminus \{0, 1\})$$$, and so on...

By naming elements in this way, David can get all the natural numbers $$$\mathbb N = \{0, 1, 2, \ldots\} \subsetneq \mathcal S$$$.

In order to go beyond, he defines $$$\infty = \min(\mathcal S \setminus \mathbb N)$$$. To get a "number" that's even bigger than $$$\infty$$$, he chooses $$$\min(\mathcal S \setminus (\mathbb N\cup\{\infty\}))$$$.

To make this idea general, David represents a "number" by a counting program P, which takes as input a (countable, possibly infinite) subset $$$T\subsetneq\mathcal S$$$. Its output, denoted by P($$$T$$$), satisfies $$$T\subseteq\text{P}(T)\subsetneq\mathcal S$$$. Counting programs belong to a context-free language:

• The empty program does nothing to $$$T$$$, returning it as-is.
• + takes the smallest unused element of $$$\mathcal S$$$ and puts it in $$$T$$$. Formally, $$$+(T) = T\cup \{\min(\mathcal S \setminus T)\}$$$.
• PQ runs the program P, followed by the program Q. Formally, $$$(\text{PQ})(T) = Q(\text{P}(T))$$$.
• [P] runs the program P in an "infinite loop". Formally, $$$[\text{P}](T) = \text{P}(T)\cup (\text{PP})(T)\cup (\text{PPP})(T)\cup\ldots$$$

Let $$$T_1 = \text{P}_1(\emptyset)$$$ and $$$T_2 = \text{P}_2(\emptyset)$$$ be the subsets constructed by the counting programs $$$\text{P}_1$$$ and $$$\text{P}_2$$$, respectively. The following conditions are equivalent, and when they hold we'll say that $$$\text{P}_1 \lt \text{P}_2$$$:

• $$$\min(\mathcal S \setminus T_1) \lt \min(\mathcal S \setminus T_2)$$$
• $$$T_1 \subsetneq T_2$$$

Thanks to his general definition, David can be mathematically precise about what he means by "infinity plus one": he means the program [+]+, which constructs the subset ([+]+)$$$(\emptyset) = \mathbb N\cup\{\infty\}$$$.

To decide who won their game, the children need you to compare numbers that are described by counting programs. Given a list of $$$M$$$ counting programs, sort them from smallest to biggest.