
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 that in the convex quadrilateral $PQRS$, $PQ = 24, \angle P = 45^\circ$, and $\overline{PQ} \parallel \overline{RS}$, the lengths of the sides, in some order, form a geometric progression, and side $\overline{PQ}$ is a side of maximum length, determine the sum of all possible values of $x$.

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