
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
Consider twelve circles of diameter 2 packed in the first quadrant of the coordinate plane, arranged in four rows, with each row containing three circles. Let region $\mathcal{Q}$ be the union of these twelve circular regions. Line $m$, with slope 2, divides $\mathcal{Q}$ 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(a, b, c) = 1. Calculate $a^2 + b^2 + c^2$.

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