
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
Find the radius of convergence of the power series with complex terms:
1) \(\sum_{n=0}^{+\infty} 2^{n}(\sqrt{n-1}-i) z^{n}\)
2) \(\sum_{n=1}^{+\infty} \frac{2^{n}}{n!}\)
3) \(\sum_{n=1}^{+\infty} \frac{(3+4 i)^{n}}{n^{2}} z^{3 n}\)

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