
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 positive real numbers \( p \) and \( q \). A positive real number sequence \(\{a_{n}\}\) satisfies:
\[ a_{0} = 1, a_{n+2} = p a_{n} - q a_{n+1} \ (n \in \mathbf{N}). \]

Find all possible values of \( a_{1} \) in terms of \( p \) and \( q \).

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