
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
Based on the following conditions, write out the first 4 terms of the sequence and conjecture the general formula for the sequence (no proof required).
1. $a_1=0$, $a_{n+1} = \frac{1}{2-a_n}$;
2. For all $n \in \mathbb{N}^*$, $a_n > 0$, and $2\sqrt{S_n} = a_n + 1$.

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