
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 that the parabola $C$ has one focus at $F\left( \frac{1}{2},0\right)$ and the corresponding directrix has the equation $x=-\frac{1}{2}$.

1. Write the equation of the parabola $C$;
2. A line passing through point $F$ intersects curve $C$ at points $A$ and $B$. Point $O$ is the coordinate origin. Find the trajectory equation of the centroid $G$ of $\triangle AOB$;
3. Point $P$ is a moving point on parabola $C$. A circle with equation $(x-3)^{2}+y^{2}=2$ is drawn through point $P$, and the tangent points are $M$ and $N$. When is $|MN|$ at its minimum? Find the minimum value of $|MN|$.

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