
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, the coordinates of point $P$ are $ \left(2m+5,3m+3\right)$.  
$(1)$ If point $P$ lies on the $x$-axis, then $m=$______.  
$(2)$ If point $P$ lies on a line passing through point $A\left(-5,1\right)$ and parallel to the $y$-axis, find the coordinates of point $P$.  
$(3)$ Point $P$ is first moved $2$ units to the right and then moved $3$ units up to obtain point $M$. If point $M$ lies in the third quadrant and the distance from point $M$ to the $y$-axis is $7$, find the coordinates of point $M.

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