
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 2 are packed in the first quadrant of the coordinate plane. Let region $\mathcal{R}$ be the union of these nine circular regions. Line $l$, with slope 4, divides $\mathcal{R}$ into two regions of equal area. Additionally, line $l$ passes through the point $\left(2,1\right)$. The equation of line $l$ can be expressed in the form $ax=by+c$, where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1. Find $a^2+b^2+c^2$.

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