
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
Let  $a,b,c$  be distinct real numbers such that  $a+b+c>0$ . Let  $M$  be the set of  $3\times 3$  matrices with the property that each line and each column contain all given numbers  $a,b,c$ . Find  $\{\max \{ \det A \mid A \in M \}$  and the number of matrices which realise the maximum value.

*Mircea Becheanu*

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