
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 the hyperbola $\frac{x^2}{1} - \frac{y^2}{3} = 1$ has an eccentricity of $\frac{m}{2}$, and the parabola $y^2 = mx$ has a focus at $F$, point $P(2,y_0) (y_0 > 0)$ is on this parabola, and $M$ is the midpoint of segment $PF$, determine the distance from point $M$ to the directrix of this parabola.

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