
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
For an invertible  $n\times n$  matrix  $M$  with integer entries we define a sequence  $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$  by the recurrence  $M_0=M$  , $M_{i+1}=(M_i^T)^{-1}M_i$  for  $i\geq 0$ .

Find the smallest integer  $n\geq 2 $   for wich there exists a normal  $n\times n$  matrix with integer entries such that its sequence  $\mathcal{S}_M$  is not constant and has period  $P=7$  i.e  $M_{i+7}=M_i$ .
( $M^T$  means the transpose of a matrix  $M$  . A square matrix is called normal if  $M^T M=M M^T$  holds).

*Proposed by Martin Niepel (Comenius University, Bratislava)..*

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