
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 two points P and Q on the hyperbola $x^2-\frac{y^2}{2}=1$ that are symmetric about the line $y=-x+b$, and the midpoint of PQ lies on the parabola $y^2=8x$. Find the value of the real number $b$.

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