
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
Define a function  $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$  by

\[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\]

(1) Let  $f_n(x)=a_n\sin x+b_n\cos x.$  Express  $a_{n+1},\ b_{n+1}$  in terms of  $a_n,\ b_n.$ (2) Find  $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$ 

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