
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 a sequence $\{a_n\}$ where each term is non-zero and the sequence is an arithmetic progression with common difference $d$, and $S_n$ represents the sum of the first $n$ terms of this sequence. It is known that $a_n^2=S_{2n-1}$ for $n \in \mathbb{N^*}$. Another sequence $\{b_n\}$ is defined such that $b_n=\frac{1}{a_n a_{n+1}}$, and $T_n$ represents the sum of the first $n$ terms of $\{b_n\}$.  
(I) Find $a_1$, $d$, and $T_n$;  
(II) If the inequality $\lambda T_n < n + 8(-1)^n$ always holds for any $n \in \mathbb{N^*}$, find the range of the real number $\lambda$.

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