
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 positive integers  $a$  such that there exists a set  $X$  of  $6$  integers satisfying the following conditions: for every  $k=1,2,\ldots ,36$  there exist  $x,y\in X$  such that  $ax+y-k$  is divisible by  $37$ .

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