
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 1 are arranged in the first quadrant of a coordinate plane. Five circles are in the base row with centers at $(0.5, 0.5)$, $(1.5, 0.5)$, $(2.5, 0.5)$, $(3.5, 0.5)$, $(4.5, 0.5)$, and the remaining five directly above the first row with centers at $(0.5, 1.5)$, $(1.5, 1.5)$, $(2.5, 1.5)$, $(3.5, 1.5)$, $(4.5, 1.5)$. Let region $\mathcal{S}$ be the union of these ten circular regions. Line $m,$ with slope $-2$, divides $\mathcal{S}$ into two regions of equal area. Line $m$'s equation can be expressed in the form $px=qy+r$, where $p, q,$ and $r$ are positive integers whose greatest common divisor is 1. Find $p^2+q^2+r^2$.

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