
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 $P_1$ be a $150$-gon whose vertices are placed in the Cartesian plane. The sum of the x-coordinates of the vertices of $P_1$ is $3024$. Using midpoints of consecutive vertices, we construct a new $150$-gon $P_2$, and similarly, by using midpoints of $P_2$, we construct another $150$-gon $P_3$. 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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