
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 sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = n^2 - 4n$, where $n \in \mathbb{N}^*$.
(I) Find the general formula for the sequence $\{a_n\}$;
(II) Let $b_n = 2^{a_n} + 1$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$;
(III) If for any positive integer $n$, $\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \dots + \frac{1}{a_na_{n+1}} \leqslant \lambda$, find the minimum value of the real number $\lambda$.

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