
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 $a_1=1$, $a_{n+1}=2a_n+1$ ($n\in\mathbb{N}^*$).
1. Find the general formula for the sequence $\{a_n\}$.
2. If the sequence $\{b_n\}$ satisfies $$4^{b_1-1}\cdot 4^{2b_2-1} \cdot 4^{3b_3-1}\cdots4^{nb_n-1}=(a_n+1)^n$$, find the general formula for the sequence $\{b_n\}$.

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