
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
Consider an equilateral triangle with a side length of 16 cm. The diameter of a circle lies along one side of this triangle. Calculate the sum of the areas of the two shaded regions, where each region is bounded by an arc and the lines connecting the triangle's vertices and a central point on the circle's diameter. In simplest radical form, the sum of the areas of the two shaded regions is $a\pi - b\sqrt{c}$. What is the value of $a + b + c$?

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