
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 ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{7} = 1$ with left and right foci $F_{1}$, $F_{2}$ coinciding with the foci of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$. Also, the line $x - y - 1 = 0$ intersects the right branch of the hyperbola at point $P$. Determine the equation of the hyperbola when its eccentricity is minimized.

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