
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 parabola $C$: $y^{2}=2px(p > 0)$ whose focus $F$ coincides with the right focus of the ellipse $\frac {x^{2}}{4}+ \frac {y^{2}}{3}=1$.  
$(1)$ Find the equation of the parabola $C$;  
$(2)$ If a line segment $AB$ of fixed length $5$ moves with its endpoints on the parabola $C$, and its midpoint is $M$, find the shortest distance from point $M$ to the $y$-axis, and the coordinates of point $M$ at this time.

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