
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 \( f(x) \) be a function satisfying the following conditions:
1. 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 \( f(z) = w - z \);
2. The equation \( f(x) = 0 \) has at least one solution, and there is a solution of this equation that is less than or equal to all other solutions;
3. \( f(0) = 1 \);
4. \( f(-2007) \leq 2008 \);
5. \( f(x) f(y) = f(x f(y) + y f(x) + x y) \).

Find the value of \( f(-2007) \).

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