
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 height of a regular triangular prism \( ABC A_1 B_1 C_1 \) is 12. A sphere \( \Omega \) with radius \( r = \frac{3\sqrt{5}}{2} \) touches all the lateral faces of the prism. Points \( T \) and \( F \) are chosen on segments \( CC_1 \) and \( BB_1 \) respectively, such that \( FT \parallel BC \), and the planes \( ABT \) and \( FA_1C_1 \) touch the sphere \( \Omega \). Find the volume of the prism and the length of segment \( CT \).

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