
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 that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.
(1) Find the equation of the trajectory $C$ of the circle's center;
(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $O$ is the coordinate origin. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$ and $| \overrightarrow{AB}|$.

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