
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 real numbers \( x, y \). For any pair of real numbers \( a_{0}, a_{1} \), define a sequence by \( a_{n+2}=x a_{n+1}+y a_{n} \) for \( n \geq 0 \). Suppose that there exists a fixed nonnegative integer \( m \) such that, for every choice of \( a_{0} \) and \( a_{1} \), the numbers \( a_{m}, a_{m+1}, a_{m+3} \), in this order, form an arithmetic progression. Find all possible values of \( y \).

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