
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
Point $P$ is an arbitrary point on the side $BC$ of an equilateral triangle $\triangle ABC$. From point $P$, lines $PX$ and $PY$ are drawn perpendicular to $AB$ and $AC$ respectively. Connect $XY$. Calculate the relationship between the perimeter of $\triangle XAY$ and the perimeter of quadrilateral $BCYX$.

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