
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
Determine the number of possible values for the determinant of  $A$ , given that  $A$  is a  $n\times n$  matrix with real entries such that  $A^3 - A^2 - 3A + 2I = 0$ , where  $I$  is the identity and  $0$  is the all-zero matrix.

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