
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
Nine circles of diameter 1 are tightly packed in the first quadrant of the coordinate plane. Let region $\mathcal{R}$ be the union of these nine circular regions. A line $l$ with slope 2 divides $\mathcal{R}$ into two equal-area parts. The equation of line $l$ can be expressed in the form $ax=by+c$, where $a, b,$ and $c$ are positive integers, with the greatest common divisor of $a, b, c$ being 1. Find $a^2+b^2+c^2.$

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