
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 \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the inequality
\[ 
f(y)-\left(\frac{z-y}{z-x} f(x)+\frac{y-x}{z-x} f(z)\right) \leq f\left(\frac{x+z}{2}\right)-\frac{f(x)+f(z)}{2} 
\]
for all real numbers \( x < y < z \).

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