
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 two sequences $\{a\_n\}$ and $\{b\_n\}$ that satisfy $a_{n+1} - a_n = 2(b_{n+1} - b_n)$ for all $n \in \mathbb{N}^*$.

1. If $a_1 = 1$ and $b_n = 3n + 5$, find the general term formula for the sequence $\{a\_n\}$.
2. If $a_1 = 6$, $b_n = 2^n$ for all $n \in \mathbb{N}^*$, and $\lambda a_n > 2^n + n + 2\lambda$ holds true for all $n \in \mathbb{N}^*$, find the range of $\lambda$.

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