
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
Let \( a_1 = 1 \), and \( a_{n+1} = \sqrt{a_n^2 - 2a_n + 2} + b \) for \( n \in \mathbb{N}^* \). 

(1) If \( b = 1 \), find \( a_2 \), \( a_3 \), and the general term of the sequence \( \{a_n\} \).

(2) If \( b = -1 \), determine whether there exists a real number \( c \) such that \( a_{2n} < c < a_{2n+1} \) holds for all \( n \in \mathbb{N}^* \).

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