
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 that the sequence $\{a_n\}$ is an arithmetic sequence with $a_3 = 5$ and a common difference $d\neq 0$, and the three terms $a_1$, $a_2$, $a_5$ form a geometric progression.
(1) Find the general formula for the sequence $\{a_n\}$ and the sum of the first $n$ terms $S_n$.
(2) If the sequence $\{b_n\}$ satisfies $b_n = \frac{1}{a_n \cdot a_{n+1}}$, and $T_n$ is the sum of the first $n$ terms of $\{b_n\}$, find $T_n$.
(3) Under the condition of (2), if the inequality $\lambda T_n < n + 8(-1)^n$ holds for all $n \in \mathbb{N}^*$, find the range of the real number $\lambda$.

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