
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 a hyperbola $T: \frac{x^2}{4} - y^2 = 1$, a line passing through point $B(-2, 0)$ intersects the hyperbola $T$ at point $A$ (where $A$ is not a vertex of the hyperbola). If the midpoint $Q$ of segment $AB$ lies on the line $y = x$, and point $P$ is any other point on the hyperbola $T$ distinct from $A$ and $B$, and is not a vertex of the hyperbola, and the lines $AP$ and $BP$ intersect the line $y = x$ at points $M$ and $N$ respectively, find the value of $\overrightarrow{OM} \cdot \overrightarrow{ON}$.

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