
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 limits of the following sequences:

a) \(\left\{a_{n}\right\}=\left\{\left(\frac{3+4 i}{7}\right)^{n}, n \in \mathbb{N}\right\}\);

b) \(\left\{a_{n}\right\}=\left\{\frac{(1+i)^{n}}{2+(1+i)^{n}}, \quad n \in \mathbb{N}\right\}\);

c) \(\left\{a_{n}\right\}=\left\{\left|\frac{z^{n}}{z^{n}}\right| \frac{(\cos i-i \sin i)^{n}}{(\cos i+i \sin i)^{n}}-2 \operatorname{ch} 2 n\right\}, z \neq 0+0 i\).

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