
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
Let  $t(A)$  denote the sum of elements of a nonempty set  $A$  of integers, and define  $t(\emptyset)=0$ . Find a set  $X$  of positive integers such that for every integers  $k$  there is a unique ordered pair of disjoint subsets  $(A_{k},B_{k})$  of  $X$  such that  $t(A_{k})-t(B_{k}) = k$ .

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