
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 its left focus $F(-c,0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point $M$ is on the ellipse in the first quadrant. Line $FM$ is intercepted by the circle $x^{2}+y^{2}= \frac{b^{2}}{4}$ with a segment of length $c$, and $|FM|= \frac{4\sqrt{3}}{3}$.
(I) Find the slope of line $FM$;
(II) Find the equation of the ellipse;
(III) If point $P$ moves on the ellipse such that the slope of line $FP$ is greater than $\sqrt{2}$, find the range of the slope of line $OP$ (where $O$ is the origin).

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