
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
Suppose the sequence $\{a_n\}$ is a geometric sequence with a common ratio less than 1 and all terms are positive. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Given that $S_2=12$, and $a_1, a_2+1, a_3$ form an arithmetic sequence.

(1) Find the general formula of the sequence $\{a_n\}$.
(2) If $b_n=a_n\cdot(n-\lambda)$, and $\{b_n\}$ is a monotonically decreasing sequence, find the range of the real number $\lambda$.

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