
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
The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what?

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