
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
Let  $n$  be a positive integer. Find the largest nonnegative real number  $f(n)$  (depending on  $n$ ) with the following property: whenever  $a_1,a_2,...,a_n$  are real numbers such that  $a_1+a_2+\cdots +a_n$  is an integer, there exists some  $i$  such that   $\left|a_i-\frac{1}{2}\right|\ge f(n)$ .

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