
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 \( z_1, z_2, z_3, z_4 \) be the solutions to the equation \( x^4 + 3x^3 + 3x^2 + 3x + 1 = 0 \). Then \( |z_1| + |z_2| + |z_3| + |z_4| \) can be written as \( \frac{a + b \sqrt{c}}{d} \), where \( c \) is a square-free positive integer, and \( a, b, d \) are positive integers with \( \operatorname{gcd}(a, b, d) = 1 \). Compute \( 1000a + 100b + 10c + d \).

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