
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
As shown in the diagram, in the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{2} + y^{2} = 1$ has left and right foci $F_{1}$ and $F_{2}$, respectively. Let $P$ be a point on $\Gamma$ in the first quadrant, and let the extended lines $P F_{1}$ and $P F_{2}$ intersect $\Gamma$ at points $Q_{1}\left(x_{1}, y_{1}\right)$ and $Q_{2}\left(x_{2}, y_{2}\right)$, respectively. Find the maximum value of $y_{1} - y_{2}$.

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