
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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