
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 circle with equation $x^{2}+y^{2}=1$, let $P$ be an arbitrary point on the circle, and draw a perpendicular line $PN$ from $P$ to the x-axis, with foot $N$. Determine the equation of the trajectory of the midpoint $M$ of line segment $PN$.

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