
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
For an ordered  $10$ -tuple of nonnegative integers  $a_1,a_2,\ldots, a_{10}$ , we denote
    \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When  $i=1$ , we take  $a_1+a_2+\cdots+a_{i-1}$  to be  $0$ . Let  $N$  be the average of  $f(a_1,a_2,\ldots,a_{10})$  over all  $10$ -tuples of nonnegative integers  $a_1,a_2,\ldots, a_{10}$  satisfying
    \[a_1+a_2+\cdots+a_{10}=20.\]
    Compute the number of positive integer divisors of  $N$ .


*2021 CCA Math Bonanza Individual Round #14*

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