
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
In $\triangle ABC$, let point $O$ be the center of its circumcircle, and $\vec{AO}=x\vec{AB}+y\vec{AC}$.

(1) If $A= \frac{2\pi}{3}$, $|\vec{AB}|=1$, $|\vec{AC}|=2$, find the values of $x$ and $y$;

(2) If $\cos A= \frac{1}{3}$, find the maximum value of $x+y$.

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