
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 $H_i$ (for $i=1,2,3,4$ and $A_5=A_1$) be the trisection points of the sides $A_i A_{i+1}$ of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ that are closer to $A_i$. Construct the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ given points $H_{1}, H_{2}, H_{3}$, and $H_{4}$.

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