
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
For a unit circle  $O$ , arrange points  $A,B,C,D$  and  $E$  in that order evenly along  $O$ 's circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as  $A', B', C', D'$  and  $E'$ , where the prime of any point is opposite the point. The length of  $AC'$  can be written as an expression  $f(x)$ , where  $f$  is a trigonometric function. Find this expression.

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