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

Determine the smallest number that Katka could have originally thought of.

(Hint: Express the sum on the fourth line using the originally thought number.)

Present the answer in LaTex format: \boxed{Your answer}
user