
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
1. Given the hyperbola \( C: (y + a)^{2} - (x - a)^{2} = 2a \), and its asymptotes pass through the point \((3, 1)\), find the equations of the asymptotes of \(C\).
2. For a hyperbola with asymptotes \( y = \pm(x + 1) \) and a focal length of 4, find its equation.

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