
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 $ABCD$ be a convex quadrilateral with a point $P$ inside it such that $P$ is equidistant from all vertices. Let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $PBC,$ $PCD,$ $PDA,$ and $PAB$ respectively. Find $\frac{[G_A G_B G_C G_D]}{[PBCD]}.$

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