
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 distance between the left vertex of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1(a > 0,b > 0)$ and the focus of the parabola $y^{2}=2px(p > 0)$ is $4$, and the coordinate of the intersection point of one asymptote of the hyperbola and the axis of symmetry of the parabola is $(-2,-1)$, find the equation of the hyperbola.

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