
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) In $\Delta ABC$, if $A=60^{\circ}$ and $a=\sqrt{3}$, then the value of $\frac{\sin A+\sin B+\sin C}{a+b+c}$ is __________.

(2) If the sums of the first $n$ terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ are $A_n$ and $B_n$ respectively, and it is given that $\frac{A_n}{B_n}=\frac{4n+2}{5n-5}$, then the value of $\frac{a_5}{b_5}$ is __________.

(3) If in the sequence $\{a_n\}$, $a_1=1$ and $a_{n+1}=3a_n+3^{n+1}$, then the general formula for the sequence is $a_n=$__________.

(4) In $\triangle ABC$, where $A$, $B$, and $C$ are the internal angles, and $a$, $b$, and $c$ are the opposite sides, given that $2\sqrt{2}(\sin^2A-\sin^2C)=(a-b)\sin B$ and the radius of the circumcircle of $\triangle ABC$ is $\sqrt{2}$, then the angle $C=$__________.

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