
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
Given the function $f(x)=\sin (\omega x+\varphi)$ $(\omega > 0,0\leqslant \varphi\leqslant \pi)$ is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\sqrt {4+\pi^{2}}$.  
$(1)$ Find the analytic expression of $f(x)$;  
$(2)$ If $\tan \alpha+ \frac {1}{\tan \alpha}=5$, find the value of $\frac { \sqrt {2}f(2\alpha- \frac {\pi}{4})-1}{1-\tan \alpha}$.

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