
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: $\tan\alpha=-\frac{1}{3}$, $\cos\beta= \frac{\sqrt{5}}{5}$, with $\alpha, \beta \in (0, \pi)$.

(1) Find the value of $\tan(\alpha+\beta)$;  
(2) Find the maximum and minimum values of the function $f(x)= \sqrt{2}\sin(x-\alpha)+\cos(x+\beta)$.

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