
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
A circle is tangent to the point $F(1,0)$ and the line $x = -1$. Let $C$ be the locus of the center of this circle. Point $A$ is on $C$ in the first quadrant with a horizontal coordinate of 4. Draw a line segment $AB$ perpendicular to the $y$-axis through point $A$, with $B$ as the foot of the perpendicular. The midpoint of $OB$ is $M$.

(1) Find the equation of the locus $C$ of the center of the circle.

(2) If a line segment $MN$ is drawn perpendicular to $FA$ through point $M$ with $N$ as the foot of the perpendicular, find the coordinates of point $N$.

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