
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; $(iii)$   $f(0) = 1$ . $(iv)$   $f(1987) \leq 1988$ . $(v)$   $f(x)f(y) = f(xf(y) + yf(x) - xy)$ .

Find  $f(1987)$ .

*Proposed by Australia.*

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