
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 a hyperbola shares the same foci with the ellipse $\dfrac{x^2}{49} + \dfrac{y^2}{24} = 1$, and has asymptotes $y = \pm \dfrac{4}{3}x$.

$(1)$ Find the equation of the hyperbola.

$(2)$ Find the equation of the line that passes through the right focus of the hyperbola and has an inclination angle of $\dfrac{\pi}{3}$.

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