
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 a square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$. The square is intersected by the line $y = \frac{x}{3}$. Determine the value of the perimeter of one of the resulting congruent quadrilaterals divided by $a$, and express your answer in simplified radical form.

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