
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
Given $\overrightarrow {m}=\left(\dfrac{1}{2}\sin x,\dfrac{\sqrt{3}}{2}\right)$, $\overrightarrow {n}=\left(\cos x,\cos ^{2}x-\dfrac{1}{2}\right)$ where $x\in \mathbb{R}$, and the function $f\left(x\right)=\overrightarrow {m}\cdot \overrightarrow {n}$.
1. Find the equation of the axis of symmetry of $f\left(x\right)$.
2. In acute triangle $\triangle ABC$, where the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $f\left(A\right)=0$, $\sin B=\dfrac{4}{5}$, and $a=\sqrt{3}$, find the value of $b$.

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