
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  $F_n$  be the nth Fibonacci number, defined by  $F_1 = F_2 = 1$  and  $F_n = F_{n-1} + F_{n-2}$  for  $n > 2$ . Let  $A_0, A_1, A_2,\cdots$  be a sequence of points on a circle of radius  $1$  such that the minor arc from  $A_{k-1}$  to  $A_k$  runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\]
for  $k \geq 1$ , where  $\mu(XY )$  denotes the radian measure of the arc  $XY$  in the clockwise direction. What is the limit of the radian measure of arc  $A_0A_n$  as  $n$  approaches infinity?

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