
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
Read the following material and answer the questions given below: Two algebraic expressions containing square roots are multiplied together. If their product does not contain square roots, we say that these two algebraic expressions are rationalized factors of each other, for example, $\sqrt{a}$ and $\sqrt{a}$, $\sqrt{2}+1$ and $\sqrt{2}-1$. When simplifying an expression with a denominator containing square roots, we can use the method of multiplying the numerator and denominator by the rationalized factor of the denominator, for example: $\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{2}×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{\sqrt{6}}{3}$, $\frac{2}{3-\sqrt{3}}=\frac{2(3+\sqrt{3})}{(3-\sqrt{3})(3+\sqrt{3})}=\frac{2(3+\sqrt{3})}{9-3}=\frac{2(3+\sqrt{3})}{6}=\frac{3+\sqrt{3}}{3}$.<br/>$(1)$ Please write down the rationalized factor of $3+\sqrt{11}$: ______;<br/>$(2)$ Please simplify $\frac{1-b}{1-\sqrt{b}}$ following the method above ($b\geqslant 0$ and $b\neq 1)$;<br/>$(3)$ Given $a=\frac{1}{\sqrt{3}-2}$, $b=\frac{1}{\sqrt{3}+2}$, find the value of $\sqrt{{a}^{2}+{b}^{2}+2}$.

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