
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
In an ellipse, we have the following conclusion: The midpoint of a chord with a slope of 1 on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ lies on the line $\frac{x}{a^{2}}+ \frac{y}{b^{2}}=0$. By analogy with the above conclusion, the correct conclusion for the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ is: The midpoint of a chord with a slope of 1 on the hyperbola lies on the line \_\_\_\_\_\_.

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