
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
Chords \( AB \) and \( CD \) of a circle \( \Gamma \) with center \( O \) have a length of 5. The extensions of segments \( BA \) and \( CD \) beyond points \( A \) and \( D \) respectively intersect at point \( P \). Line \( PO \) intersects segment \( AC \) at point \( L \), and \( AL:LC = 1:2 \).

a) Find \( AP \).

b) Additionally, let the radius of circle \( \Gamma \) be 6.5 and let point \( T \) be the center of the circle inscribed in triangle \( ACP \). Find the length of segment \( PT \) and the area of triangle \( ACP \).

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