
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
Suppose that  $n\ge3$  is a natural number. Find the maximum value  $k$  such that there are real numbers  $a_1,a_2,...,a_n \in [0,1)$  (not necessarily distinct) that for every natural number like  $j \le k$  , sum of some  $a_i$ -s is  $j$ .

*Proposed by Navid Safaei*

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