
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
A polynomial with integer coefficients is called a "multivariate", if all its complex roots are distinct Gaussian integers (complex numbers whose real and imaginary parts are both integers). Find the number of such polynomials that meet the following criteria:

\[ f(z) = z^{n} + c_{n-1} z^{n-1} + \cdots + c_{2} z^{2} + 3 c_{1} z + 50 \]

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