
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
Consider the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) which satisfy:
\[
a_1 = 3, \quad b_1 = 1
\]
and for any \(n \in \mathbf{Z}_{+}\), 
\[
\begin{cases}
a_{n+1} = a_n + b_n + \sqrt{a_n^2 - a_n b_n + b_n^2}, \\
b_{n+1} = a_n + b_n - \sqrt{a_n^2 - a_n b_n + b_n^2}.
\end{cases}
\]

1. Find the general formulas for the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\).

2. Let \([x]\) denote the greatest integer not exceeding the real number \(x\). Define
\[ S_n = \sum_{i=1}^{n} [a_i], \quad T_n = \sum_{i=1}^{n} [b_i]. \]
Find the smallest \(n \in \mathbf{Z}_{+}\) such that
\[
\sum_{k=1}^{n} (S_k + T_k) > 2017.
\]

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