
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
Given the functions \( y=2^x \), \( y=\log_2 x \), \( y=x^2 \), and \( y=\cos 2x \), determine the number of these functions that satisfy the inequality \( f\left(\frac{x_1 + x_2}{2}\right) > \frac{f(x_1) + f(x_2)}{2} \) for all \( 0 < x_1 < x_2 < 1 \).

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