
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with a minor axis length of $2$ and an eccentricity of $\dfrac{\sqrt{2}}{2}$, the line $l$: $y=kx+m (k \neq 0)$ intersects the ellipse $C$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ passes through the point $(0, -\dfrac{1}{2})$.
1. Find the standard equation of the ellipse $C$;
2. When the area of $\triangle AOB$ ($O$ is the origin) reaches its maximum value, find the equation of the line $l$.

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