
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 the sequence  $\{y_n\}_{n=1}^{\infty}$  defined by  $y_1=y_2=1$  and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers  $k$  such that every term of the sequence is a perfect square.

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