
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
The set \( A \) consists of functions \( f(x) \) that satisfy the following property: For any \( x > 0 \) and \( y > 0 \), and \( x \neq y \), the inequality \( f(x) + 2f(y) > 3f\left(\frac{x + 2y}{3}\right) \) holds. Determine whether the following functions belong to the set \( A \):
(1) \( f_{1}(x) = \log_{2} x \)
(2) \( f_{2}(x) = (x + 1)^{2} \)

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