
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
Points $A$ and $B$ are the left and right endpoints of the major axis of ellipse $C$: $\frac{x^{2}}{36}+ \frac{y^{2}}{20}=1$, respectively. Point $F$ is the right focus of the ellipse. Point $P$ is on the ellipse and is located above the $x$-axis, with $PA$ perpendicular to $PF$.

(1) Find the coordinates of point $P$.
(2) Let $M$ be a moving point on ellipse $C$. Find the trajectory equation of the midpoint $G$ of line segment $MF$.

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