
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 \( f \) be a function that satisfies the following conditions:
(i) If \( x > y \) and \( f(y) - y \geq v \geq f(x) - x \), then \( f(z) = v + z \) for some number \( z \) between \( x \) and \( y \).
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions.
(iii) \( f(0) = 1 \).
(iv) \( f(1987) \leq 1988 \).
(v) \( f(x) f(y) = f(x f(y) + y f(x) - x y) \).

Find \( f(1987) \).

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