
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
Find all strictly increasing sequences of positive integers  $a_1, a_2, \ldots$  with  $a_1=1$ , satisfying   $$ 3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n} $$   for all positive integers  $n$ .

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