
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  $m$  be a positive integer and let  $A$ , respectively  $B$ , be two alphabets with  $m$ , respectively  $2m$  letters. Let also  $n$  be an even integer which is at least  $2m$ . Let  $a_n$  be the number of words  of length  $n$ , formed with letters from  $A$ , in which appear all the letters from  $A$ , each an even number of times. Let  $b_n$  be the number of words of length  $n$ , formed with letters from  $B$ , in which appear all the letters from  $B$ , each an odd number of times. Compute  $\frac{b_n}{a_n}$ .

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