
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 $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, and $a\_1 = 3$. If the sequence $\{S\_n + 1\}$ is a geometric sequence with a common ratio of 4:

1. Find $S\_n$ and the general term formula $a\_n$ for the sequence $\{a\_n\}$.
2. Let $b\_n = n \cdot 4^n + λa\_n$, $n ∈ ℕ^*$. If the sequence $\{b\_n\}$ is increasing, find the range of the real number $λ$.

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