
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 $M(1,m)$ on the parabola $C:y^{2}=2px\left(p \gt 0\right)$ such that the distance between $M$ and the focus is $2$.  
$(1)$ Find $p$ and $m$;  
$(2)$ If there exist points $A$ and $B$ on the parabola $C$ such that $MA\bot MB$, let $D$ be the midpoint of $AB$, and the distance from $D$ to the directrix of the parabola $C$ is $\frac{15}{2}$, find the coordinates of point $D$.

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