
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 $\{a\_n\}$ whose product of the first $n$ terms is $T\_n$, and $\log_{\sqrt{2}}{T\_n}=n(n-1)$, $n\in\mathbb{N}^*$.

(1) Find the general term formula of the sequence $\{a\_n\}$;

(2) Given $b\_n=ka\_n-2 (n\in\mathbb{N}^*)$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $S\_n$. If the sequence $\{S\_n\}$ is a strictly increasing sequence for any $n\in\mathbb{N}^*$, find the range of the real number $k$.

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