
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
It is known that the sequence $\{a_n\}$ consists of positive integers and satisfies $a_{n+1} = a_n^2 - 2na_n + 2$ for $n \in \mathbb{N}$. Also, $a_5 = 11$.

(Ⅰ) Find the values of $a_1$, $a_2$, $a_3$, $a_4$, and from this, conjecture the general formula for $\{a_n\}$ (proof not required);

(Ⅱ) Let $b_n = 11 - a_n$, $S_n = b_1 + b_2 + \ldots + b_n$, $S_n' = |b_1| + |b_2| + \ldots + |b_n|$, find the value of $$\lim_{n\rightarrow \infty } \frac {S_{n}}{S_{n}'}$$;

(Ⅲ) Let $C_n = \frac {1}{n(1+a_{n})}$ for $n \in \mathbb{N}$, $T_n = C_1 + C_2 + \ldots + C_n$, is there a maximum integer $m$ such that for any $n \in \mathbb{N}$, $T_n > \frac {m}{32}$? If it exists, find the value of $m$; if not, explain why.

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