
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 moving circle $M$ passes through a fixed point $E(2,0)$, and cuts a chord $PQ$ of length $4$ on the $y$-axis.

$(1)$ Find the equation of the trajectory $C$ of the center $M$ of the circle;
$(2)$ Let $A$, $B$ be two points on the trajectory $C$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=-4$, $F(1,0)$, denote $S=S_{\triangle OFA}+S_{\triangle OAB}$, find the minimum value of $S$.

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