
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
Camille the snail lives on the surface of a regular dodecahedron. Right now, he is on vertex \( P_1 \) of the face with vertices \( P_1, P_2, P_3, P_4, P_5 \). This face has a perimeter of 5. Camille wants to get to the point on the dodecahedron farthest away from \( P_1 \). To do so, he must travel along the surface a distance at least \( L \). What is \( L^2 \)?

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