
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
Right $\triangle PQR$ has sides $PQ = 5$, $QR = 12$, and $PR = 13$. Rectangle $ABCD$ is inscribed in $\triangle PQR$ such that $A$ and $B$ are on $\overline{PR}$, $D$ on $\overline{PQ}$, and $C$ on $\overline{QR}$. If the height of the rectangle (parallel to side $\overline{PQ}$) is half its length (parallel to side $\overline{PR}$), find the length of the rectangle.

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