
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 $f(x)=a^{x}$ where $a>0$ and $a\neq 1$ passes through the point $(2, \frac{1}{9})$.  
$(1)$ Find the value of $a$.  
$(2)$ Compare $f(2)$ and $f(b^{2}+2)$.  
$(3)$ Find the range of the function $f(x)={a}^{{x}^{2}-2x}$ for $x\geqslant 0$.

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