
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 $\triangle ABC$, $AB=\sqrt{3}$, $AC=1$, $\angle A=30^{\circ}$, find the area of $\triangle ABC$.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $(\sin A-\sin B)(a+b)=(\frac{1}{2} a-c)\sin C$, find $\cos B$.
3. Given that $\{a_{n}\}$ is a geometric sequence with positive terms and $a_{1}\cdot a_{7}=\frac{3}{4}$, find $a_{4}$.
4. In the sequence $\{a_{n}\}$, $a_{1}=1$ and $a_{n+1}=a_{n}+2$ ($n\in\mathbb{N}^{*}$), find $a_{20}$.

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