
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passing through points $(\sqrt{2}, 0)$ and $(0,1)$, with its right focus being $F$.
(I) Find the equation of the ellipse $C$;
(II) A line $l$ passing through point $F$ intersects the ellipse $C$ at points $A$ and $B$. If $\overrightarrow{AF}=3\overrightarrow{FB}$, find the value of $| \overrightarrow{OA}+ \overrightarrow{OB}|$ (where $O$ is the coordinate origin).

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