
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 the function $f\left( x \right)=a\cos x+2b{\cos ^{2}}x+1-b$.

(1) When $b=1,a=1$, find the range of the function $f\left( x \right)$;

(2) If $b=1$, find the range of real number $a$ such that the equation $\left| f\left( x \right) \right|\geqslant {{a}^{2}}$ holds for some real number $x$.

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