Katka thought of a five-digit natural number. On the first line of her notebook, she wrote the sum of the thought number and half of the thought number. On the second line, she wrote the sum of the thought number and one fifth of the thought number. On the third line, she wrote the sum of the thought number and one ninth of the thought number. Finally, she added all three numbers she had written and wrote the result on the fourth line. To her surprise, she found that the number on the fourth line was the cube of a certain natural number.

Determine the smallest number Katka could have thought of at the beginning.

(L. Růžičková)

Hint. Express the sum on the fourth line in terms of the originally thought number.