
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 $C: y^2 = 2px$ ($p>0$), with focus $F$, and directrix $l$. A point $A$ on the parabola $C$ has an $x$-coordinate of 3, and the distance from point $A$ to the directrix $l$ is 5.

(1) Find the equation of the parabola $C$.
(2) If $P$ is a moving point on the parabola $C$, find the equation of the trajectory for the midpoint $M$ of the line segment $FP$.

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