
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 triangle $DEF$, the sides are given by $DE = 25$, $EF = 31$, and $FD = 34$. Points $P$ and $Q$ are on $\overline{DF}$ with $P$ on $\overline{DQ}$, points $R$ and $S$ are on $\overline{DE}$ with $R$ on $\overline{SE}$, and points $T$ and $U$ are on $\overline{EF}$ with $T$ on $\overline{UF}$. In addition, the points are positioned so that $\overline{PQ}\parallel\overline{EF}$, $\overline{RS}\parallel\overline{FD}$, and $\overline{TU}\parallel\overline{DE}$. Right angle folds are then made along $\overline{PQ}$, $\overline{RS}$, and $\overline{TU}$. Determine the maximum possible height (denoted as $h$) of a table constructed from triangle $DEF$ whose top is parallel to the floor, and express $h$ in the form $\frac{k\sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime.

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