
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 $x \in \mathbb{R}, k \in \mathbb{R}$.

1. If $x \in \left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]$ 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 $f(x)$.

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