
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
Ten circles each with diameter 1 are packed in the first quadrant of the coordinate plane, aligned in a rectangular grid where the rows contain, from top to bottom, 4, 3, and 3 circles respectively. Let region $\mathcal{A}$ be the union of these ten circular regions. Line $m,$ with a slope of 2, divides $\mathcal{A}$ into two regions of equal area. The equation of line $m$ 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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