
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
Consider the ellipse given by the equation $\frac{x^2}{12} + \frac{y^2}{3} = 1$. One of its foci is $F$, and $P$ is a point on the ellipse. If the midpoint $M$ of the line segment $PF$ lies on the $y$-axis, then the $y$-coordinate of point $M$ is $\boxed{\pm \frac{\sqrt{3}}{4}}$.

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