
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 $\{a_n\}$ where $a_1=1$, $a_3=9$, and $a_n=a_{n-1}+\lambda n-1$ (for $n\geqslant 2$).

(I) Find the value of $\lambda$ and the general formula for the sequence $\{a_n\}$.

(II) Let $b_n=(-1)^n\cdot (a_n+n)$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$. Find $S_{2n}$.

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