
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
Find all prime numbers  $p$  such that  $p^3$  divides the determinant
\[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 11 & 3^2 & 1 & \dots & 1 1 & 1 & 4^2 & & 1 \vdots & \vdots & & \ddots & 1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]

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