
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 chords \( AB \) and \( CD \) of a circle \( \Gamma \) with center \( O \) have a length of 8. The extensions of line segments \( BA \) and \( CD \) beyond points \( A \) and \( D \) intersect at point \( P \). The line \( PO \) intersects segment \( AC \) at point \( L \), and \( AL : LC = 1 : 5 \).

a) Find \( AP \).

b) Additionally, assume the radius of the circle \( \Gamma \) is 5, and point \( T \) is 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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