
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 the function $f(x)= \sqrt{3}\sin^2\left(x+ \frac{\pi}{4}\right)-\cos^2x- \frac{1+ \sqrt{3}}{2}$ $(x\in\mathbb{R})$.

(1) Find the minimum value of the function $f(x)$ and its smallest positive period;

(2) If $A$ is an acute angle, and the vector $m=(1,5)$ is perpendicular to the vector $n=(1,f\left( \frac{\pi}{4}-A\right))$, find the value of $\cos 2A$.

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