
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
Let $PQRS$ be a quadrilateral where $\overline{PQ} = \overline{RS}$, and $\overline{PQ}$ is not parallel to $\overline{SR}$. The diagonals intersect at point $O$ such that $PO = 30$ and $RO = 15$. Suppose the angle $\angle POS = \frac{\pi}{4}$. Let $T$ be the midpoint of $\overline{PQ}$, and $M$ be the foot of the perpendicular from $Q$ to $\overline{SR}$. Find the distance $TM$.

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