
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 $\overrightarrow{a} = (\sqrt{3}\cos x, \cos x)$, $\overrightarrow{b} = (0, \sin x)$, $\overrightarrow{c} = (\sin x, \cos x)$, and $\overrightarrow{d} = (\sin x, \sin x)$.

(1) When $x = \frac{\pi}{4}$, find the angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$;

(2) Find the value of $x$ when $\overrightarrow{c} \cdot \overrightarrow{d}$ reaches its maximum value;

(3) Let the function $f(x) = (\overrightarrow{a} - \overrightarrow{b}) \cdot (\overrightarrow{c} + \overrightarrow{d})$. The graph of function $f(x)$ is translated right by $s$ units and up by $t$ units (where $s, t > 0$) to obtain the graph of function $g(x)$, and $g(x) = 2\sin 2x + 1$. Let $\overrightarrow{m} = (s, t)$, find the minimum value of $|\overrightarrow{m}|$.

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