
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 \(\boldsymbol{m}=\left(\cos \frac{x}{2},-1\right)\) and \(\boldsymbol{n}=\left(\sqrt{3} \sin \frac{x}{2}, \cos ^{2} \frac{x}{2}\right)\), let 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 angles \(A\), \(B\), and \(C\) be \(a\), \(b\), and \(c\), respectively, and satisfy \(2b \cos A \leq 2c - \sqrt{3}a\). Find the range of \(f(B)\).

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