
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 $\{a_n\}$ that satisfies $a_{n+1}-a_n=2[f(n+1)-f(n)]$ $(n\in\mathbb{N}^*)$.  

(1) If $a_1=1$ and $f(x)=3x+5$, find the general formula for the sequence $\{a_n\}$.  

(2) If $a_1=6$, $f(x)=2^x$, and $\lambda a_n > 2^n+n+2\lambda$ holds for all $n\in\mathbb{N}^*$, find the range of the real number $\lambda$.

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