
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
Let  $ABCDEFGH$  be a regular octagon with side length  $\sqrt{60}$ . Let  $\mathcal{K}$  denote the locus of all points  $K$  such that the circumcircles (possibly degenerate) of triangles  $HAK$  and  $DCK$  are tangent. Find the area of the region that  $\mathcal{K}$  encloses.

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