
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
Find the trajectory equation of the moving point in the following scenarios:
1. Let C be a circle: $(x-1)^2 + y^2 = 1$. Through the origin O, draw any chord of the circle and find the trajectory equation of the midpoint of the chord.
2. In the Cartesian plane $xOy$, the distance of point M to point F(1,0) is 1 unit more than its distance to the y-axis. Denote the trajectory of point M as C. Find the equation of trajectory C.

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