
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
Take  $k\in \mathbb{Z}_{\ge 1}$  and the sets  $A_1,A_2,\dots, A_k$  consisting of  $x_1,x_2,\dots ,x_k$  positive integers, respectively. For any two sets  $A$  and  $B$ , define  $A+B=\{a+b~|~a\in A,~b\in B\}$ .

Find the least and greatest number of elements the set  $A_1+A_2+\dots +A_k$  may have.

*(Andrei Bâra)*

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