
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
Find all functions  $f: \mathbb{R}\to\mathbb{R}$  satisfying:

 $\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}$   for all  $x,y,z \in \mathbb{R}$ 

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