
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 $\{a_n\}$ with partial sum $S_n$ satisfying $S_n = n^2$, there exists another sequence $\{b_n\}$ defined by $b_n = \frac{1}{a_n a_{n+1}}$, and $T_n$ is the partial sum of the first $n$ terms of sequence $\{b_n\}$.  
$(I)$ Find the general term formula $a_n$ and the partial sum $T_n$ for the sequence $\{b_n\}$.  
$(II)$ If for any $n \in N^*$ the inequality $λT_n < n + (-1)^n$ holds, find the range of the real number $λ$.

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