
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
(1) If the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=\frac{2}{3}a_n+\frac{1}{3}$, then the general formula for $\{a_n\}$ is $a_n=\_\_\_\_\_\_\_\_$.

(2) In $\triangle ABC$, the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=2$, $B=\frac{\pi}{6}$, and $c=2\sqrt{3}$, then $b=\_\_\_\_\_\_\_\_$.

(3) Let the lengths of the sides opposite angles $A$, $B$, and $C$ in $\triangle ABC$ be $a$, $b$, and $c$, respectively. If $b+c=2a$ and $3\sin A=5\sin B$, then angle $C=\_\_\_\_\_\_\_\_$.

(4) In a positive geometric sequence $\{a_n\}$, $a_5=\frac{1}{2}$, and $a_6+a_7=3$. The maximum positive integer value of $n$ for which $a_1+a_2+\ldots+a_n > a_1a_2\ldots a_n$ is $\_\_\_\_\_\_\_\_$.

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