
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
Given a triangle $PQR$ with sides $PQ = 20$, $QR = 21$, and $RP = 29$. Points $E$ and $F$ are on $\overline{QR}$ with $E$ on $\overline{PF}$, and point $G$ is on $\overline{RP}$ with $G$ on $\overline{PE}$. Lines $\overline{EF}$ and $\overline{PG}$ are constructed such that $\overline{EF} \parallel \overline{PQ}$ and $\overline{PG} \parallel \overline{QR}$. Determine the maximum height of a structure that can be built by folding along these lines such that the top of the structure remains parallel to the base $PQ$.

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