
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
The sequence \(\left\{y_{n}\right\}\) is defined as follows:
\[
y_{1} = y_{2} = 1, \quad y_{n+2} = (4k-5)y_{n+1} - y_{n} + 4 - 2k, \quad n = 1, 2, \ldots
\]
Find all integers \(k\) such that every term in the sequence \(\left\{y_{n}\right\}\) is a perfect square.

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