
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
A polygon $Q$ in the Cartesian plane has $39$ vertices. The sum of the $x$-coordinates of these vertices equals $117$. The midpoints of the sides of polygon $Q$ form a new $39$-gon, $R$. Then, the midpoints of the sides of $R$ form another $39$-gon, $S$. If the midpoints of the sides of $S$ form another $39$-gon, $T$, find the sum of the $x$-coordinates of the vertices of $T$.

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