The Horror of the Nested Sum

A nested sum is one in which one summation is inside another. Let the function f(m,n)f(m,n) be equal to the nested sum a1=1m(a2=1a1(a3=1a2((an1=1an2(an=1an1an))))).\displaystyle \sum_{a_1=1}^{m} \left(\displaystyle \sum_{a_2=1}^{a_1} \left(\displaystyle \sum_{a_3=1}^{a_2} \left(\cdots \left(\displaystyle \sum_{a_{n-1}=1}^{a_{n-2}} \left(\displaystyle \sum_{a_n=1}^{a_{n-1}} a_n\right)\right)\cdots\right)\right)\right). For how many values of n1000n\leq 1000 will f(m,n)f(m+1,n)f(m,n)|f(m+1,n) if m=11m=11?

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