
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 vectors $\overrightarrow {m}=(\sqrt {3}\sin \dfrac{x}{4}, 1)$ and $\overrightarrow {n}=(\cos \dfrac{x}{4}, \cos ^{2}\dfrac{x}{4})$, and the function $f\left(x\right)=\overrightarrow {m}\cdot \overrightarrow {n}$.
1. If $f\left(x\right)=1$, find the value of $\cos \left(\dfrac{2\pi }{3}-x\right)$.
2. In $\triangle ABC$, where the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and satisfy $a\cos C+\dfrac{1}{2}c=b$, find the range of values of $f\left(B\right)$.

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