
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
Triangle $PQR$ has side lengths $PQ=150, QR=250$, and $PR=200$. Lines $\ell_P, \ell_Q$, and $\ell_R$ are drawn parallel to $\overline{QR}, \overline{PR}$, and $\overline{PQ}$, respectively, such that the intersections of $\ell_P, \ell_Q$, and $\ell_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 50$, and $25$, respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_P, \ell_Q$, and $\ell_R$.

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