
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 hyperbola $x^2-\frac{y^2}{4}=1$ and the line $l:y=kx+m\left(k\neq \pm 2\right)$ have a unique common point $M$, where the line passing through point $M$ and perpendicular to $l$ intersects the $x$-axis and $y$-axis at points $A\left(x,0\right)$ and $B\left(0,y\right)$, respectively. <br/>$(1)$ When $m=2$ and $k \gt 0$, find the coordinates of point $M$; <br/>$(2)$ As point $M$ moves, find the trajectory equation of point $P\left(x,y\right)$.

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