
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
Five lighthouses are located, in order, at points  $A, B, C, D$ , and  $E$  along the shore of a circular lake with a diameter of  $10$  miles. Segments  $AD$  and  $BE$  are diameters of the circle. At night, when sitting at  $A$ , the lights from  $B, C, D$ , and  $E$  appear to be equally spaced along the horizon. The perimeter in miles of pentagon  $ABCDE$  can be written  $m +\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $m + n$ .

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