
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 integer  $n$ , let  $I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos (2n+1)x}{\sin x}\  dx.$ (1) Find  $I_0.$ (2) For each positive integer  $n$ , find  $I_n-I_{n-1}.$ (3) Find  $I_5$ .

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