
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
Given a set of vectors $ \overrightarrow{a\_1}, \overrightarrow{a\_2}, \overrightarrow{a\_3}, …, \overrightarrow{a\_n} (n \in \mathbb{N}^*) $, let $ \overrightarrow{S\_n} = \overrightarrow{a\_1} + \overrightarrow{a\_2} + \overrightarrow{a\_3} + … + \overrightarrow{a\_n} $. If there exists $ \overrightarrow{a\_p} (p \in \{1, 2, 3, …, n\}) $ such that $ |\overrightarrow{a\_p}| \geqslant |\overrightarrow{S\_n} - \overrightarrow{a\_p}| $, then $ \overrightarrow{a\_p} $ is called the "$h$ vector" of the vector set.

1. If $ \overrightarrow{a\_n} = (n, x + n) (n \in \mathbb{N}^*) $, and $ \overrightarrow{a\_3} $ is the "$h$ vector" of the vector set $ \overrightarrow{a\_1}, \overrightarrow{a\_2}, \overrightarrow{a\_3} $, find the range of values for the real number $x$.
2. If $ \overrightarrow{a\_n} = (\frac{1}{3}^{n-1} \cdot (-1)^{n} (n \in \mathbb{N}^*) $, does the vector set $ \overrightarrow{a\_1}, \overrightarrow{a\_2}, \overrightarrow{a\_3}, …, \overrightarrow{a\_n} $ have an "$h$ vector"? Provide your conclusion and explain your reasoning.
3. Given that $ \overrightarrow{a\_1}, \overrightarrow{a\_2}, \overrightarrow{a\_3} $ are all "$h$ vectors" of the vector set $ \overrightarrow{a\_1}, \overrightarrow{a\_2}, \overrightarrow{a\_3} $, where $ \overrightarrow{a\_1} = (\sin x, \cos x) $, $ \overrightarrow{a\_2} = (2\cos x, 2\sin x) $. In the plane rectangular coordinate system, there is a sequence of points $Q\_1, Q\_2, Q\_3, …, Q\_n$ such that: $Q\_1$ is the coordinate origin, $Q\_2$ is the terminal point of the position vector $ \overrightarrow{a\_3} $, and $Q\_{2k+1}$ and $Q\_{2k}$ are symmetric about point $Q\_1$, $Q\_{2k+2}$ and $Q\_{2k+1} (k \in \mathbb{N}^*)$ are symmetric about point $Q\_2$. Find the minimum value of $| \overrightarrow{Q\_{2013}Q\_{2014}} |$.

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