
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and a minor axis length of $2$. $F\_1$, $F\_2$ are the left and right foci, and $O$ is the coordinate origin.
(1) Find the standard equation of the ellipse;
(2) Circle $O$ has diameter $F\_1$, $F\_2$, and line $l$: $y = kx + m$ is tangent to circle $O$ and intersects the ellipse at points $A$ and $B$. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = \frac{2}{3}$, find the value of $k$.

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