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

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