
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
A function $f(x)$ is called a "Γ-function" if for a certain real number pair $(a, b)$, the equation $f(a+x) \cdot f(a-x) = b$ holds for all $x$. 
1. Determine whether the functions $f_1(x) = x$ and $f_{2}(x) = 3^{x}$ are "Γ-functions".
2. If $f_{3}(x) = \tan(x)$ is a "Γ-function", find all ordered real number pairs $(a, b)$ that satisfy the conditions.
3. If a function $f(x)$ defined on $\mathbb{R}$ is a "Γ-function", and there exist real number pairs $(0, 1)$ and $(1, 4)$ that satisfy the conditions, and when $x \in [0, 1]$, the range of $f(x)$ is $[1, 2]$, find the range of $f(x)$ when $x \in [-2016, 2016]$.

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