
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
Given the plane region defined by $\begin{cases}\begin{matrix}x\geqslant 0, \\ y\geqslant 0,\end{matrix} \\ x+2y-4\leqslant 0\end{cases}$ is exactly covered by the smallest circle $C$: $(x-a)^{2}+(y-b)^{2}=r^{2}$ and its interior, then the equation of circle $C$ is __________.

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