
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 greater than two. What are the largest value \( h \) and the smallest value \( H \) such that

$$
h < \frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\ldots+\frac{a_{n}}{a_{n}+a_{1}} < H
$$

for any positive numbers \( a_{1}, a_{2}, \ldots, a_{n} \)?

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