
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, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the orthocenters of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively. Find $\frac{[H_A H_B H_C H_D]}{[ABCD]}.$

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