
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 functions $f\left(x\right)=a\cdot 4^{x}-a\cdot 2^{x+1}+1-b\left(a \gt 0\right)$ and $g\left(x\right)=k\cdot 2^{x}$, where the maximum value of function $f\left(x\right)$ on the interval $\left[1,2\right]$ is $9$ and the minimum value is $1$. The graphs of functions $f\left(x\right)$ and $g\left(x\right)$ have two distinct intersection points on the interval $\left[-1,2\right]$. Find the possible values of the real number $k$.

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