
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 with a focal length of $2$ and equation $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ ($a > 0, b > 0$), there exists a point $P$ on the hyperbola such that the ratio of its distances to the $y$-axis and $x$-axis is $2$. Determine the range of values for the length of the hyperbola's transverse axis.

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