
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 the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) be defined as follows:
\[ a_{1} = 3, \quad b_{1} = 1 \]

and for any \( n \in \mathbb{Z}_{+} \), we have
\[
\begin{cases}
a_{n+1} = a_{n} + b_{n} + \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}, \\
b_{n+1} = a_{n} + b_{n} - \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}.
\end{cases}
\]

(1) Find the general terms of the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\).

(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define \(S_{n} = \sum_{i=1}^{n} \left[a_{i}\right]\) and \(T_{n} = \sum_{i=1}^{n} \left[b_{i}\right]\). Find the smallest \(n \in \mathbb{Z}_{+}\) such that
\[
\sum_{k=1}^{n} \left( S_{k} + T_{k} \right) > 2017.
\]

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