
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 a point $M(2m+1, m-4)$ in the Cartesian coordinate system.<br/>$(1)$ If point $N(5,2)$ and $MN$ is parallel to the $x$-axis, find the coordinates of $M$.<br/>$(2)$ If point $M$ is to the right of the $y$-axis and the distance to the $y$-axis is $3$, find the coordinates of $M.

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