
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
Consider a function $f(x)$ that satisfies: there exists an $x_0$ in its domain $D$ such that for a given constant $t$, the equation $f(x_0 + t) = f(x_0) \cdot f(t)$ holds. Such $f(x)$ is called a $t$-level distribution function on its domain. Investigate the following problems:
(1) Determine whether the functions $f(x) = 2x$ and $g(x) = \frac{2}{x}$ are 1-level distribution functions, and provide the reasoning.
(2) Determine whether the function $\varphi(x)= \sqrt{\frac{a}{x^2+1}}$ (where $a>0$) can be a 2-level distribution function, and if possible, find the range of the parameter $a$; If not, explain the reason.
(3) Discuss whether there exists a real number $a$ such that for any constant $t$ (where $t \in \mathbb{R}$), the function $\varphi(x)= \sqrt{\frac{a}{x^2+1}}$ (with $a > 0$) is a $t$-level distribution function on its domain. If such an $a$ exists, determine its range; if not, explain the reason.

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