
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
If a sequence $\{a_n\}$ satisfies the conditions:  
(1) There exist distinct $p$, $q \in \mathbb{N}^*$ such that $a_p = a_q = c$ ($c$ is a constant);  
(2) When $n \neq p$ and $n \neq q$, for any $n \in \mathbb{N}^*$, we have $a_n > c$, then the sequence $\{a_n\}$ is called a double-bottom sequence.  
(1) Determine whether the following sequences $\{a_n\}$ are double-bottom sequences (only conclusions are required without proof):  
(i) $a_n = n + \frac{6}{n}$; (ii) $a_n = \sin \frac{n\pi}{2}$;    (iii) $a_n = |(n-3)(n-5)|$;  
(2) Let $a_n = \begin{cases}101-2n, & 1 \leqslant n \leqslant 50 \\ 2^{n-50}+m, & n > 50\end{cases}$, if the sequence $\{a_n\}$ is a double-bottom sequence, find the value of the real number $m$ and the sum of the first $n$ terms of the sequence $S_n$;  
(3) Let $a_n = (kn+3)\left( \frac{9}{10} \right)^n$, is there an integer $k$ such that the sequence $\{a_n\}$ is a double-bottom sequence? If so, find all values of $k$; if not, explain why.

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