
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}
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