
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 altitudes $A A^{\prime}, B B^{\prime}, C C^{\prime}$ of the triangle $A B C$ intersect at point $H$. Let $P$ be the midpoint of the segment $A H$. What is the condition for the quadrilateral $A^{\prime} B^{\prime} P C^{\prime}$ to be a parallelogram?


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