
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 $\{a\_n\}$ where all terms are positive, the sum of the first $n$ terms is $S\_n$, and $S\_n=a\_n^2+ \frac{1}{2}a\_n$, $n∈\mathbb{N}^∗$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. Suppose the sequence $\{b\_n\}$ satisfies: $b\_1=1$, $b\_n-b\_{n-1}=2a\_n\ (n≥slant 2)$, find the sum of the first $n$ terms of the sequence $\{ \frac{1}{b\_n}\}$, denoted as $T\_n$.
3. If $T\_n≤λ(n+4)$ holds for any $n∈\mathbb{N}^∗$, find the range of $λ$.

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