
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
Let \( f: (-1, +\infty) \to (-1, +\infty) \) be a continuous and monotonic function with \( f(0) = 0 \) and satisfying the inequality
$$
f[x + f(y) + x f(y)] \geqslant y + f(x) + y f(x) \quad \text{for all} \quad x, y \in (-1, +\infty).
$$
Determine the function \( f(x) \).

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