
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 of diameter 2 are packed in the first quadrant of the coordinate plane, arranged as three rows with the first and second rows containing four circles each and the third row containing two circles. Let region $\mathcal{R}$ be the union of these ten circular regions. Line $l$, with slope 5, divides $\mathcal{R}$ into two regions of equal area. Line $l$'s equation 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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