
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 equations of the asymptotes of a hyperbola are $y = 3x + 4$ and $y = -3x + 6$. Given that the hyperbola passes through the point $(1, 9)$, find the standard form for the equation of the hyperbola: \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\] where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find $a - h.$

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