
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
In the Cartesian coordinate system, if the initial side of angle $\alpha$ coincides with the non-negative half-axis of the $x$-axis, and the terminal side passes through the point $P(-1,2)$.  
$(1)$ Find the values of $\sin \alpha$ and $\tan \alpha$;  
$(2)$ If $f(\alpha)=\frac{{\sin({\frac{\pi}{2}+\alpha})\tan({\pi+\alpha})+2\cos({\pi-\alpha})}}{{\sin\alpha+\cos({-\alpha})}}$, simplify and find the value.

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