
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 center of a hyperbola is at the origin and one of its foci is $F(\sqrt{7},0)$, the line $y=x-1$ intersects it at points $M$ and $N$. If the x-coordinate of the midpoint of $MN$ is $-\frac{2}{3}$, then the equation of this hyperbola is \_\_\_\_\_\_.

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