
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
A sequence of integers \( a_1, a_2, a_3, \ldots \) is defined by
\[
\begin{array}{c}
a_1 = k, \\
a_{n+1} = a_n + 8n \text{ for all integers } n \geq 1.
\end{array}
\]

Find all values of \( k \) such that every term in the sequence is a square.

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