
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 a circle $C: (x-a)^2 + (y-b)^2 = 1$, let the planar region $\Omega = \{ (x,y) | \begin{cases} x+y-7 \leq 0 \\ x-y+3 \geq 0 \\ y \geq 0 \end{cases} \}$, if the center of the circle $C \in \Omega$ and the circle $C$ is tangent to the $x$-axis, determine the maximum value of $a^2 + b^2$.

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