
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  $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$  For constant  $a$  such that  $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$ , evaluate  $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$ 

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