
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 $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, its right focus point lies on the line $x+y-\sqrt{3}=0$, which intersects $M$ at points $A$ and $B$. Let $P$ be the midpoint of $AB$ such that the slope of $OP$ is $\frac{1}{2}$. Determine the equation of ellipse $M$.

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