
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  $n$  be a positive integer with  $d$  digits, all different from zero. For  $k = 0,. . . , d - 1$ , we define  $n_k$  as the number obtained by moving the last  $k$  digits of  $n$  to the beginning. For example, if  $n = 2184$  then  $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$ . For  $m$  a positive integer, define  $s_m(n)$  as the number of values  $k$  ​​such that  $n_k$  is a multiple of  $m.$  Finally,  define  $a_d$  as the number of integers  $n$  with  $d$  digits all nonzero, for which  $s_2 (n) + s_3 (n) + s_5 (n) = 2d.$   
Find  \[\lim_{d \to \infty} \frac{a_d}{5^d}.\]

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