
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 $x,k \in \mathbb{R}$,
1. If $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ and $\overrightarrow{a} \parallel (\overrightarrow{b}+\overrightarrow{c})$, find the value of $x$.
2. If the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$, find the minimum value of $f(x)$.
3. Does there exist a real number $k$ such that $(\overrightarrow{a}+\overrightarrow{d}) \perp (\overrightarrow{b}+\overrightarrow{c})$? If so, find the range of values for $k$. If not, explain why.

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