
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively.
(1) If $a=3c$, $b=\sqrt{2}$, and $\cos B=\frac{2}{3}$, find the value of $c$.
(2) If $\frac{\sin A}{a}=\frac{\cos B}{2b}$, find the value of $\sin(B+\frac{\pi}{2})$.

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