
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  $n \geq 3$ , let  $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$  Let  $C_n = 
(c_{i, j})$  the  $n \times n$  matrix defined by  $c_{i, j} = b _{(j -i) \mod n}$ . Show 
that  $\det (C_n) = 3$  if  $n$  is not a multiple of 3 and  $\det (C_n) = 0$  if  $n$  
is a multiple of 3.

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