
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 an ellipse $M$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity $e= \frac{\sqrt{3}}{2}$, and a point $(\sqrt{2}, \frac{1}{2})$ on the ellipse.
(1) Find the standard equation of the ellipse $M$;
(2) A line $l$ with a slope of $1$ intersects the ellipse $M$ at two distinct points $A$ and $B$. If a circle passing through the origin $O$ has $AB$ as its diameter, find the equation of the line $l$.

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