
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
Point \( P \) is selected at random from the interior of the hexagon with vertices \( A = (0, 0) \), \( B = (3, 0) \), \( C = (4.5, \sqrt{3}) \), \( D = (3, 2\sqrt{3}) \), \( E = (0, 2\sqrt{3}) \), and \( F = (-1.5, \sqrt{3}) \). What is the probability that \( \angle APB \) is obtuse?

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