
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 curve C, any point P on curve C is at a distance that is 3 units less than its distance to the line m: x = -4 from point F (1, 0).
(1) Find the equation of curve C;
(2) Does there exist a point M on curve C such that the sum of the distances from M to F (1, 0) and from M to A (3, 2) is minimized? If such a point exists, find the minimum value and the coordinates of M; if not, explain why.

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