
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 two fractions  $a/b$  and  $c/d$  we define their *pirate sum* as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$  where the two initial fractions are simplified the most possible, like the result.
For example, the pirate sum of  $2/7$  and  $4/5$  is  $1/2$ .
Given an integer  $n \ge 3$ , initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$ .
At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction.
Determine, in function of  $n$ , the maximum and the minimum possible value for the last fraction.

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