
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 infinite sequences  $a_1, a_2, \ldots$  of positive integers satisfying the following properties:
(a)  $a_1 < a_2 < a_3 < \cdots$ ,
(b) there are no positive integers  $i$ ,  $j$ ,  $k$ , not necessarily distinct, such that  $a_i+a_j=a_k$ ,
(c) there are infinitely many  $k$  such that  $a_k = 2k-1$ .

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