
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
Given a positive integer \( n \), find the smallest value of \(\left\lfloor\frac{a_{1}}{1}\right\rfloor + \left\lfloor\frac{a_{2}}{2}\right\rfloor + \cdots + \left\lfloor\frac{a_{n}}{n}\right\rfloor\) over all permutations \((a_{1}, a_{2}, \ldots, a_{n})\) of \((1,2, \ldots, n)\).

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