
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 the sequence $\{a\_n\}$, where $a\_1=1$, $a\_3=9$, and $a\_n=a\_{n-1}+λn-1$ for $n\geqslant 2$.

(I) Find the value of $λ$ and the general term formula for the sequence $\{a\_n\}$.

(II) Let $b\_n=(-1)^{n}\cdot (a\_n+n)$, and $S\_{2n}$ denotes the sum of the first $2n$ terms of the sequence $\{b\_n\}$. Find $S\_{2n}$.

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