
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
Let \( n \in \mathbf{N}_{+} \), and \((\sqrt{2}+\sqrt{3})^{2n-1} = a_{n} \sqrt{2} + b_{n} \sqrt{3} \). If \( a_{n+1} = p a_{n} + q b_{n} \), then \( p+q = \quad \) , and the value of \( 2a_{n}^{2} - 3b_{n}^{2} \) is \quad .

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