
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

...

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system
a) We call *admissible sequence* a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences.

b) For each integer  $ n\geq 2$  we denote  $ d_n$  the number of possibilities of completing with even digits an array with  $ n$  rows and 4 columns, such that 

(1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array.

Find the values of  $ n$  for which the number  $ \frac {d_{n\plus{}1}}{d_n}$  is an integer.

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