
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 $P_1$ with $50$ vertices is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $50$ vertices equals $1005$. The midpoints of the sides of $P_1$ form a second polygon $P_2$. The midpoints of the sides of $P_2$ form a third polygon $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$.

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