
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 vectors $\overrightarrow{a} = (2 + \sin x, 1)$, $\overrightarrow{b} = (2, -2)$, $\overrightarrow{c} = (\sin x - 3, 1)$, and $\overrightarrow{d} = (1, k)$ (where $x \in \mathbb{R}, k \in \mathbb{R}$),
(I) If $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\overrightarrow{a}$ is parallel to ($\overrightarrow{b} + \overrightarrow{c}$), find the value of $x$;
(II) Does there exist a real number $k$ and $x$ such that ($\overrightarrow{a} + \overrightarrow{d}$) is perpendicular to ($\overrightarrow{b} + \overrightarrow{c}$)? If it exists, find the range of values for $k$; if not, provide a reason.

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