
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 an interger  $n\geq 2$ , determine the maximum value the sum  $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$  may achieve, and the points at which the maximum is achieved, as  $a_1,a_2,...a_n$  run over all positive real numers subject to  $a_k\geq a_1+a_2...+a_{k-1}$ , for  $k=2,...n$ 

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