Description:
Jellyfish is playing a one-player card game called "Slay the Spire". There are $$$n$$$ cards in total numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th card has power $$$c_i$$$.
There is a binary string $$$s$$$ of length $$$n$$$. If $$$s_i = \texttt{0}$$$, the $$$i$$$-th card is initially in the draw pile. If $$$s_i = \texttt{1}$$$, the $$$i$$$-th card is initially in Jellyfish's hand.
Jellyfish will repeat the following process until either her hand or the draw pile is empty.
1. Let $$$x$$$ be the power of the card with the largest power in her hand.
2. Place a single card with power $$$x$$$ back into the draw pile.
3. Randomly draw $$$x$$$ cards from the draw pile. All subsets of $$$x$$$ cards from the draw pile have an equal chance of being drawn. If there are fewer than $$$x$$$ cards in the draw pile, Jellyfish will draw all cards.
At the end of this process, find the probability that Jellyfish can empty the draw pile, modulo $$$1\,000\,000\,007$$$.
Formally, let $$$M=1\,000\,000\,007$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output such an integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.
Input Format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1\leq t\leq 100$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 120$$$) — the number of cards.
The second line of each test case contains $$$n$$$ integers $$$c_1,c_2,\ldots,c_n$$$ ($$$0 \leq c_i \leq n$$$) — the powers of the cards. It is guaranteed that $$$c_1 \leq c_2 \leq \ldots \leq c_n$$$.
The third line of each test case contains a binary string $$$s$$$ of length $$$n$$$. If $$$s_i = \texttt{0}$$$, the $$$i$$$-th card is initially in the draw pile. If $$$s_i = \texttt{1}$$$, the $$$i$$$-th card is initially in Jellyfish's hand.
It is guaranteed that the sum of $$$n^2$$$ over all test cases does not exceed $$$120^2$$$.
Output Format:
For each test case, output the probability that Jellyfish can empty the draw pile modulo $$$1\,000\,000\,007$$$.
Note:
In the first test case, Jellyfish will keep playing cards with power $$$1$$$ until Jellyfish draws a card with power $$$0$$$ or power $$$2$$$. If Jellyfish draws a card with power $$$0$$$, she will eventually empty her hand. If Jellyfish draws a card with power $$$2$$$, she will eventually empty the draw pile. Since there is an equal chance of drawing $$$0$$$ or $$$2$$$, the answer is $$$\frac{1}{2}$$$, and $$$2 \cdot 500\,000\,004 \equiv 1 \pmod {10^9+7}$$$