
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
Suppose nine circles of radius 1 are centered at every integer coordinate point within the square defined by vertices (0,0), (0,3), (3,0), and (3,3) in the first quadrant of the coordinate plane. Let $\mathcal{S}$ be the union of these circular regions. Line $m$, with slope 2, divides region $\mathcal{S}$ into two regions of equal area. Find the equation of line $m$ in the form $ax = by + c$, where $a, b,$ and $c$ are positive integers with gcd 1, and compute the sum $a^2 + b^2 + c^2$.

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