
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 an arithmetic sequence $\{a_n\}$ where each term is non-zero and its sum of the first $n$ terms is $S_n$. It satisfies $2S_n = a_n^2 + \lambda n, \lambda \in \mathbb{R}$.  
$(1)$ Find the value of $\lambda$;  
$(2)$ Find the sum of the first $n$ terms, $T_n$, of the sequence $\left\{ \frac{1}{a_{2n-1}a_{2n+1}} \right\}$.

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