
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 $z\_n = x\_n + i \cdot y\_n$, where $x\_n, y\_n \in \mathbb{R}$, $n \in \mathbb{N}^*$, $i$ is the imaginary unit, $z\_{n+1} = (1+i) \cdot z\_n$, $z\_1 = 3 + 4i$. The point corresponding to the complex number $z\_n$ on the complex plane is denoted as $Z\_n$.
(1) Find the values of the complex numbers $z\_2$, $z\_3$, $z\_4$;
(2) Does there exist a positive integer $n$ such that $\overrightarrow{OZ\_n} // \overrightarrow{OZ\_1}$? If so, find all $n$ that satisfy the condition; if not, explain the reason;
(3) Find the sum of the first 102 terms of the sequence ${x\_n \cdot y\_n}$.

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