
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 $xOy$, angle $a$ has $Ox$ as its initial side, and its terminal side intersects the unit circle at point $P(m,n)$ in the second quadrant.<br/>$(1)$ If $n=\frac{12}{13}$, find the values of $\tan \alpha$ and $\frac{2\sin(\pi+\alpha)+\cos\alpha}{\cos(\frac{\pi}{2}+\alpha)+2\cos\alpha}$;<br/>$(2)$ If $\sin \alpha +\cos \alpha =\frac{1}{5}$, find the coordinates of point $P$.

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