
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 regular 16-gon is inscribed in a circle of radius 16. The sum of the lengths of all sides and certain selected diagonals of the 16-gon can be written in the form
\[ a + b \sqrt{2} + c \sqrt{4}, \]
where \( a \), \( b \), and \( c \) are positive integers. Find \( a+b+c \).

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