
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
Suppose $f:[0,+\infty) \rightarrow(-\infty,+\infty)$ satisfies:
1. $f(2)=0$;
2. When $0 \leqslant x<2$, $f(x) \leqslant 0$;
3. For all $x, y>0$, the inequality $f[x f(y)] f(y) \leqslant f(x+y)$ holds.

Find $f(x)$ when $x \in[0, 2)$.

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