
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 a line $l$: $y=kx+1 (k \neq 0)$ intersects with an ellipse $3x^{2}+y^{2}=a$ at two distinct points $A$ and $B$. Point $C$ is the intersection of line $l$ and the $y$-axis.
(I) If $k=1$ and $|AB|= \frac{\sqrt{10}}{2}$, find the value of the real number $a$.
(II) If $\overrightarrow{AC}=2\overrightarrow{CB}$, find the maximum area of $\triangle AOB$ and the equation of the ellipse at this time.

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