
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 point $M(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$ on the circle $x^2 + y^2 = 1$, find the tangent line $l$ to the circle at point $M$. The intersection of tangent line $l$ and the x-axis is the focus of the parabola $E: y^2 = 2px (p > 0)$. Let $A$ and $B$ be the intersection points of line $l$ and parabola $E$. Determine the distance between the midpoint of $AB$ and the directrix of parabola $E$.

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