
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 the function \( f(x) \) satisfy the following conditions:
(i) If \( x > y \), and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \), such that \( f(z) = w - z \);
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there exists one solution that is not greater than all other solutions;
(iii) \( f(0) = 1 \);
(iv) \( f(-2003) \leq 2004 \);
(v) \( f(x) \cdot f(y) = f[x \cdot f(y) + y \cdot f(x) + xy] \).

Find the value of \( f(-2003) \).

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