
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 the circle $O: x^2+y^2=1$ with the point $O$ at the origin, and a straight line $l: y=kx+b$ ($b>0$) that is tangent to circle $O$ and intersects the ellipse $\frac{x^2}{2}+y^2=1$ at two distinct points $A$ and $B$.
(1) Let $b=f(k)$, find the expression for $f(k)$.
(2) If $\overrightarrow{OA} \cdot \overrightarrow{OB}=\frac{2}{3}$, find the equation of line $l$.
(3) If $\overrightarrow{OA} \cdot \overrightarrow{OB}=m$ ($\frac{2}{3}\leq m\leq \frac{3}{4}$), find the range of areas for triangle $OAB$.

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