
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
In the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, point $P$ is inside the ellipse $\Gamma$ and moves on the line $y=x$. Points $K$ and $L$ are on $\Gamma$, such that the directions of $\overrightarrow{P K}$ and $\overrightarrow{P L}$ are along the positive direction of the $x$-axis and $y$-axis respectively. Point $Q$ is such that $P K Q L$ forms a rectangle. Does there exist two points $A$ and $B$ in the plane such that when the rectangle $P K Q L$ changes, $|Q A|-|Q B|$ is a non-zero constant? If such points exist, find all possible lengths of line segment $A B$; if not, explain why.

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