
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
The model of a regular quadrilateral pyramid has all faces made of uniform plates of the same material (and have the same thickness). The lateral edges of the pyramid are twice as long as the sides of the base. Determine the position of the center of mass of the pyramid, given the position vectors of the apex $S$ and the four vertices of the base $A_{1}, A_{2}, A_{3}, A_{4}$.

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