
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
Let $F_1$ and $F_2$ be the two foci of the hyperbola $x^2 - y^2 = 1$, and let $O$ be the origin. Circle $O$ has diameter $F_1F_2$; line $l: y = kx + b$ is tangent to circle $O$ and intersects the hyperbola at points $A$ and $B$. The projection of vector $\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|}$ in the direction of vector $\overrightarrow{F_{1}F_{2}}$ is $p$.
(1) Find the relationship between $b$ and $k$ based on the given conditions;
(2) Determine the equation of line $l$ when $(\overrightarrow{OA} \cdot \overrightarrow{OB})p^2 = 1$;
(3) When $(\overrightarrow{OA} \cdot \overrightarrow{OB})p^2 = m$ and $2 \leq m \leq 4$, find the range of the area of triangle $AOB$.

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