
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 that the vertex of angle $\alpha$ coincides with the origin, its initial side coincides with the positive half-axis of the x-axis, and there is a point P(3,1) on its terminal side, with $\alpha \in (0, \pi)$, $\beta \in (0, \pi)$, and $\tan(\alpha-\beta) = \frac{\sin 2\left(\frac{\pi}{2}-\alpha\right) + 4\cos^2\alpha}{10\cos^2\alpha + \cos\left(\frac{3\pi}{2}-2\alpha\right)}$.  
(1) Find the value of $\tan(\alpha-\beta)$.  
(2) Find the value of $\tan \beta$.  
(3) Find the value of $2\alpha-\beta$.

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