
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
Solve the following questions:
(1) Given $\cos(\alpha+\beta)= \frac{1}{3}$ and $\cos(\alpha-\beta)= \frac{1}{5}$, find the value of $\tan\alpha\tan\beta$;
(2) Given $\theta \in \left[0, \frac{\pi}{4}\right]$, and $\sin^4\theta + \cos^4\theta = \frac{5}{8}$, find the value of $\sin\theta\cos\theta$.

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