
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 function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to \( x \).

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