
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
In the regular triangular prism \(ABC-A_1B_1C_1\), \(AB = 3\) and \(AA_1 = 4\). Let \(M\) be the midpoint of \(AA_1\), and \(P\) be a point on \(BC\). The shortest path from \(P\) along the lateral surface of the prism via edge \(CC_1\) to \(M\) has a length of \(\sqrt{29}\). Let \(N\) be the point where this shortest path intersects \(OC_1\). Determine the lengths of \(PC\) and \(NC\).

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