A positive integer \(n\) satisfies:

- \(n = (a_1)^2 + (a_2)^2\)
- \(n = (b_1)^3 + (b_2)^3 + (b_3)^3\)
- \(n = (c_1)^4 + (c_2)^4 + (c_3)^4 + (c_4)^4\)
- \(n = (d_1)^5 + (d_2)^5 + (d_3)^5 + (d_4)^5 + (d_5)^5\)

where all the variables \(n, a_1, a_2, b_1, b_2, b_3, c_1, c_2, c_3, c_4, d_1, d_2, d_3, d_4, d_5\) are distinct positive integers.

Find the smallest such \(n\).

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