
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
Consider the plane regions represented by the system of inequalities \begin{cases} y \leqslant x, \\\\ 3y \geqslant x, \\\\ x+y \leqslant 4 \end{cases} and the inequality ${(x+2)^{2}}+(y-2)^{2}\leqslant 2$. Let $\Omega_1$ and $\Omega_2$ denote these regions, respectively. For any point $M$ in $\Omega_1$ and any point $N$ in $\Omega_2$, find the minimum value of $|MN|$.

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