
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
Set $A$ is composed of functions $f(x)$ that satisfy the following property: for any two distinct real numbers $x\_1$, $x\_2$ in the domain, $\frac{1}{2}[f(x\_1) + f(x\_2)] > f(\frac{x\_1 + x\_2}{2})$.
(1) Determine whether $f(x) = x^2$ and $g(x) = \log\_2 x$ belong to set $A$ and explain the reasons;
(2) Given that $f(x) \in A$ with domain $(0, +\infty)$, range $(0, 1)$, and $f(1) > \frac{1}{2}$, find an analytic expression for such a function $f(x)$.

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