
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 positive sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, it satisfies $a_n^2 = S_n + S_{n-1}$ (for $n \geq 2$), and $a_1 = 1$.
1. Find the general formula for the sequence $\{a_n\}$.
2. Let $b_n = (1-a_n)^2 - a(1-a_n)$, if $b_{n+1} > b_n$ holds for any $n \in \mathbb{N}^*$, find the range of the real number $a$.

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