
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
$$
\boldsymbol{m}=\left(\cos \frac{x}{2},-1\right), \boldsymbol{n}=\left(\sqrt{3} \sin \frac{x}{2}, \cos ^{2} \frac{x}{2}\right) ,
$$

define the function $f(x)=\boldsymbol{m} \cdot \boldsymbol{n}+1$.
1. If $x \in\left[0, \frac{\pi}{2}\right]$ and $f(x)=\frac{11}{10}$, find the value of $\cos x$.
2. In $\triangle ABC$, let the sides opposite to $\angle A$, $\angle B$, and $\angle C$ be $a$, $b$, and $c$, respectively. If $2 b \cos A \leqslant 2 c-\sqrt{3} a$, find the range of values for $f(B)$.

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