
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
Given the real coefficient equation \(x^{3} + 2(k-1)x^{2} + 9x + 5(k-1) = 0\) has a complex root with a modulus of \(\sqrt{5}\), find the value of \(k\) and solve the equation.

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