
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 5. A sphere \(\Omega\) with radius \(r = \frac{\sqrt{33}}{4}\) is tangent to all the lateral faces of the prism. Points \(F\) and \(L\) are chosen on the segments \(CC_{1}\) and \(BB_{1}\), respectively, such that \(FL \parallel BC\), and the planes \(LAC\) and \(FA_{1}B_{1}\) are tangent to the sphere \(\Omega\). Find the volume of the prism and the length of the segment \(CF\).

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