
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
 A circle is inscribed in a regular octagon with area  $2024$ . A second regular octagon is inscribed in the circle, and its area can be expressed as  $a + b\sqrt{c}$ , where  $a, b, c$  are integers and  $c$  is square-free. Compute  $a + b + c$ .


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