
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 Cartesian coordinate system $xOy$, the hyperbola $C$: $x^{2}- \frac{y^{2}}{b^{2}}=1$ ($b > 0$) has two asymptotes that intersect with the circle $O$: $x^{2}+y^{2}=2$ at four points sequentially named $A$, $B$, $C$, and $D$. If the area of rectangle $ABCD$ is $b$, then the value of $b$ is.

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