
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
The distance from a focus of the hyperbola $C: x^2 - \frac{y^2}{b^2} = 1$ $(b > 0)$ to one of its asymptotes is $1$. Find the value of $b$; if hyperbola $C_1$ is different from $C$, but has the same asymptotes as $C$, provide a possible equation for $C_1$.

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