
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 a Cartesian coordinate system, the vertex of angle $\alpha$ coincides with the origin, the initial side of angle $\alpha$ coincides with the positive half of the $x$-axis, and the terminal side intersects the graph of the linear function $y=-x+1$ at point $P(m,n)$.<br/>$(1)$ When $m=-2$, find the value of $\sin \alpha \cdot \cos \alpha$;<br/>$(2)$ If $\frac{sin(\frac{π}{2}+\alpha)cos(π-\alpha)}{tan(\frac{3π}{2}-\alpha)cot(π+\alpha)}=-\frac{1}{4}$, find the coordinates of point $P$.

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