
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 that the line $y = x + 1$ intersects the ellipse $mx^2 + ny^2 = 1$ (where $m > n > 0$) at points A and B, and the midpoint of chord AB has an x-coordinate of $-\frac{1}{3}$, find the tangent of the angle between the two asymptotes of the hyperbola $\frac{x^2}{m^2} - \frac{y^2}{n^2} = 1$.

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