
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
There is a line segment of length \( l \) (where \( l \geqslant 1 \)), with both of its endpoints moving along the parabola \( y = x^2 \). Determine:
(1) The equation of the trajectory of the midpoint \( M \) of this line segment.
(2) The coordinates of the point \( M \) which is at the lowest distance from the \( x \)-axis.

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