
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
Let $a$ be a constant in $\mathbb{R}$, and the function $f(x) = 4^x - a\cdot2^{x+1} + 1$ for $x \in [1,2]$.
1. When $a=2$, find the range of the function $g(x) = \frac{1}{f(x)}$.
2. If the minimum value of the function $f(x)$ is $0$, find the value of $a$.

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