
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 $DEF$, side lengths are $DE = 25$, $EF = 29$, and $FD = 32$. Points $P$ and $Q$ are on $\overline{DE}$ so that $\overline{PQ}\parallel\overline{EF}$ and $PQ = \frac{1}{3}DE$. Points $R$ and $S$ are on $\overline{EF}$ such that $\overline{RS}\parallel\overline{FD}$ and $RS = \frac{1}{3}EF$. Points $T$ and $U$ are on $\overline{FD}$ with $\overline{TU}\parallel\overline{DE}$ and $TU = \frac{1}{3}FD$. Determine the length of segment $PQ$.

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