
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 floor plan of a castle wall is a regular pentagon with a side length of \( a (= 100 \text{ m}) \). The castle is patrolled by 3 guards along paths from which every point can see the base shape of the castle wall at angles of \( 90^\circ, 60^\circ, \) and \( 54^\circ \) respectively. Calculate the lengths of the paths, as well as the area between the innermost and outermost paths.

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