
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 \( y > 0 \), with \( x \neq y \),
\[
f(x) + 2f(y) > 3f\left( \frac{x + 2y}{3} \right).
\]

Determine whether \( f_{1}(x) = \log_{2} x \) and \( f_{2}(x) = (x + 1)^{2} \) are in the set \( A \). Provide reasoning.

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