
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
Consider a cube \( ABCD EFGH \), where \( ABCD \) and \( EFGH \) are faces, and segments \( AE, BF, CG, DH \) are edges of the cube. Let \( P \) be the center of face \( EFGH \), and let \( O \) be the center of the cube. Given that \( AG = 1 \), determine the area of triangle \( AOP \).

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