
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 \( 1 \times 1 \times 1 \) cubes, another is one \( 3 \times 2 \times 2 \) solid, another is three \( 2 \times 2 \times 1 \) solids, another is three \( 4 \times 1 \times 1 \) solids, etc. We do not consider, say, \( 2 \times 2 \times 1 \) and \( 1 \times 2 \times 2 \) 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 \( 10^{m}<B\left(2015^{100}\right)<10^{m+1} \).

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