
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let the plane region determined by the inequality ${{x}^{2}}+{{y}^{2}}\leqslant 4$ be $U$, and the plane region determined by $\left| x \right|+\left| y \right|\leqslant 1$ be $V$.

$(1)$ Define points with integer coordinates as "integer points". Randomly select $3$ integer points within region $U$. Calculate the probability that exactly $2$ of these integer points are within region $V$.

$(2)$ Let set $A=\left\{ x\in \mathbb{Z}\left| {{x}^{2}} +{{y}^{2}}=4 \right.\right\}$; set $B=\{x\in \mathbb{Z}|\left|x\right|+|y|\leqslant 1\}$. If it is possible to establish $m$ different mappings from set $A$ to set $B$, and $n$ different mappings from set $B$ to set $A$, find the values of $m$ and $n$.

Present the answer in LaTex format: \boxed{Your answer}
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