
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
The sequences \(\{a_n\}, \{b_n\}, \{c_n\}\) satisfy the initial conditions \(a_1 = 1\), \(b_1 = 3\), \(c_1 = 2\), and the recurrence relations:
\[
\begin{cases}
a_{i+1} = a_i + c_i - b_i + 2 \\
b_{i+1} = \frac{1}{2}(3c_i - a_i + 5) \\
c_{i+1} = 2a_i + 2b_i - 3 \\
(i = 1, 2, \cdots)
\end{cases}
\]

Find the sequences \(\{a_n\}, \{b_n\}, \{c_n\}\).

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