
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
1. Find the standard equation of the hyperbola that shares the same asymptotes with the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ and passes through the point $(3, 4)$.
2. The line $x + y - \sqrt{3} = 0$ passes through the right focus of the ellipse $M: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ and intersects $M$ at points $A$ and $B$. $O$ is the coordinate origin, $P$ is the midpoint of $AB$, and the slope of $OP$ is $\frac{1}{2}$. Find the equation of the ellipse $M$.

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