
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
In rectangle $ABCD$, let $P$ be a point on $BC$ such that $BP = 2PC$. Let $E$ be on $BC$ such that $BE = EP$. Also, $\angle APD = 90^{\circ}$. $R$ is the midpoint of $AD$. Point $Q$ lies on $PR$ such that $PQ = QR$. Given $PA = 25$, $PQ = 20$, and $AQ = 15$, find the length of $PD$.

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