
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
We say that a sequence  $a_1,a_2,\cdots$  is *expansive* if for all positive integers  $j,\; i<j$  implies  $|a_i-a_j|\ge \tfrac 1j$ . Find all positive real numbers  $C$  for which one can find an expansive sequence in the interval  $[0,C]$ .

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