\[\displaystyle \prod_{ i=1 }^n \text{lcm} \left(1,2,3, \cdots, \left \lfloor \frac n i \right \rfloor \right) = k! \]

If \(n=5699\), find \(k\).

**Notations:**

- \(\text{lcm}(\cdot) \) denotes the lowest common multiple function.
- \(\lfloor{\cdot}\rfloor \) denotes the floor function.

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