
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 coordinate origin, point $P(-1, \frac{\sqrt{2}}{2})$ is on the ellipse, and $\overrightarrow{PF_1} \cdot \overrightarrow{F_1F_2} = 0$, $O$ is a circle with diameter $(F_1F_2)$, line $l: y = kx + m$ is tangent to the circle and intersects the ellipse at two distinct points $A$ and $B$.
(I) Find the standard equation of the ellipse;
(II) When $\overrightarrow{OA} \cdot \overrightarrow{OB} = \lambda$, and $\frac{3}{4} \leqslant \lambda \leqslant \frac{4}{5}$, find the range of lengths of chord $AB$.

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