
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 one focus of the parabola C is F($$\frac {1}{2}$$,0), and the equation of the directrix is x=-$$\frac {1}{2}$$.
(1) Write the equation of the parabola C;
(2) (This part is only for science students) A line passing through point F intersects curve C at points A and B. Point O is the origin. Find the trajectory equation of the centroid G of △AOB;
(3) Point P is a moving point on the parabola C. A tangent line is drawn through point P to the circle (x-3)<sup>2</sup>+y<sup>2</sup>\=2, with tangent points M and N. When is the value of |MN| the smallest? Also, find the smallest value of |MN|.

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