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

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