
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
We are given  $n$  identical cubes, each of size  $1\times 1\times 1$ .  We arrange all of these  $n$  cubes to produce one or more congruent rectangular solids, and let  $B(n)$  be the number of ways to do this.  

For example, if  $n=12$ , then one arrangement is twelve  $1\times1\times1$  cubes, another is one  $3\times 2\times2$  solid, another is three  $2\times 2\times1$  solids, another is three  $4\times1\times1$  solids, etc. We do not consider, say,  $2\times2\times1$  and  $1\times2\times2$  to be different; these solids are congruent. You may wish to verify, for example, that  $B(12) =11$ .

Find, with proof, the integer  $m$  such that  $10^m<B(2015^{100})<10^{m+1}$ .

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