
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 \(ABC A_1B_1C_1\) is 12. A sphere \(\Omega\) with radius \(r=\sqrt{\frac{35}{3}}\) touches all the lateral faces of the prism. Points \(K\) and \(L\) are chosen on segments \(AA_1\) and \(BB_1\) respectively such that \(KL \parallel AB\), and the planes \(KBC\) and \(L A_1C_1\) touch the sphere \(\Omega\). Find the volume of the prism and the length of the segment \(AK\).

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