
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

Real numbers \(x, y, z\) satisfy

\[
x + xy + xyz = 1, \quad y + yz + xyz = 2, \quad z + xz + xyz = 4 .
\]

The largest possible value of \(xyz\) is \(\frac{a + b\sqrt{c}}{d}\), where \(a, b, c, d\) are integers, \(d\) is positive, \(c\) is square-free, and \(\operatorname{gcd}(a, b, d) = 1\). Find \(1000a + 100b + 10c + d\).

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