
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 sum of the first $n$ terms is $S_n$, the line $y=x-2\sqrt{2}$ intersects the circle $x^2+y^2=2a_n+2$ at points $A_n$ and $B_n$ ($n\in\mathbb{N}^*$), and $S_n= \frac{1}{4}|A_nB_n|^2$. If $a_1+2a_2+3a_3+\ldots+na_n<\lambda a_n^2+2$ holds for any $n\in\mathbb{N}^*$, determine the range of the real number $\lambda$.

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