
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 a function $f(x)$, if there exists a pair of real numbers $(a, b)$ such that the equation $f(a+x) \cdot f(a-x) = b$ holds for every $x$ in the domain of the function, then the function $f(x)$ is called an "$(a, b)$ type function".
1. Determine whether the function $f(x) = 4^x$ is an "$(a, b)$ type function" and explain your reasoning.
2. It is known that function $g(x)$ is a "$(1, 4)$ type function", and when $x \in [0, 1]$, $g(x) = x^2 - m(x - 1) + 1 (m > 0)$. If $1 \leq g(x) \leq 3$ holds for all $x \in [0, 2]$, find the range of values for $m$.

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