
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
Call a positive integer \( n \) quixotic if the value of
$$
\operatorname{lcm}(1,2,3, \ldots, n) \cdot \left(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\right)
$$
is divisible by 45. Compute the tenth smallest quixotic integer.

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