
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 $\overrightarrow{a} = (\sin x, \cos x), \overrightarrow{b} = (\sin x, k), \overrightarrow{c} = (-2\cos x, \sin x - k)$.
(1) When $x \in \left[0, \dfrac{\pi}{4}\right]$, find the range of $|\overrightarrow{b} + \overrightarrow{c}|$;
(2) If $g(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{c}$, find the value of $k$ for which the minimum value of $g(x)$ is $-\dfrac{3}{2}$.

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