
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
Fill in the blanks:

(1) In $\triangle ABC$, $A=60^{\circ}$, $b = 1$, and the area of $\triangle ABC$ is $\sqrt{3}$. Find the value of $\dfrac{a+b+c}{\sin A+\sin B+\sin C}$.

(2) In an arithmetic sequence with a common difference not equal to $0$, ${a_1}+{a_3}=8$, and $a_4$ is the geometric mean of $a_2$ and $a_9$. Find $a_5$.

(3) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $\sqrt{3}\sin A-a\cos B-2a=0$, find $\angle B$.

(4) Given a sequence $\{a_n\}$ where ${a_1}=-60$ and $a_{n+1}=a_n+3$, find the sum $|a_1|+|a_2|+|a_3|+\ldots+|a_{30}|$.

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