
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
【Full score: 14 points】  
Given the function $f(x) = \sin(x + \frac{\pi}{4}) + \cos(x - \frac{\pi}{4})$, where $x \in \mathbb{R}$.  
(1) Find the smallest positive period and the minimum value of $f(x)$.  
(2) Given $\cos(\beta - \alpha) = \frac{1}{2}$ and $\cos(\beta + \alpha) = -\frac{1}{2}$, where $0 < \alpha < \beta \leq \frac{\pi}{2}$, find the value of $f(\beta)$.

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