
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 a regular triangular prism \( ABC-A_1B_1C_1 \), \( AB = 3 \), \( AA_1 = 4 \). \( M \) is the midpoint of \( AA_1 \), and \( P \) is a point on \( BC \). The shortest path from \( P \) along the side faces of the prism, passing through edge \( CC_1 \), to \( M \) is \(\sqrt{29}\). Let \( N \) be the intersection point of this path with \( OC_1 \). Find the lengths of \( PC \) and \( NC \).

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