
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 that $F_1$ and $F_2$ are the two foci of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ ($a > b > 0$), $O$ is the origin, and point $P\left(-1, \frac{\sqrt{2}}{2}\right)$ lies on the ellipse and satisfies $\overrightarrow{PF_1} \cdot \overrightarrow{F_1F_2} =0$. Circle $⊙O$ is the circle with diameter $F_1F_2$, and line $l$: $y = kx + m$ is tangent to $⊙O$ and intersects the ellipse at two distinct points, $A$ and $B$.

(1) Find the standard equation of the ellipse.

(2) When $\overrightarrow{OA} \cdot \overrightarrow{OB} = \frac{2}{3}$, find the value of $k$.

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