
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]


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Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=2$ and $a_{2}=3$, and the sum of the first $n$ terms $S_{n}$ satisfies $S_{n+1}+S_{n-1}=2S_{n}+1$, where $\left(n\geqslant 2,n\in N^{*}\right)$.

(1) Find the general formula for the sequence $\left\{a_{n}\right\}$.

(2) Let $b_{n}=4^{n}+\left(-1\right)^{n-1}\lambda \cdot 2^{a_{n}}(\lambda $ is a non-zero integer, $n\in N^{*})$. Determine the value of $\lambda$ such that $b_{n+1} \gt b_{n}$ holds for any $n\in N^{*}$.

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