
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  $k$  be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where  $\lfloor x\rfloor$  denotes the greatest integer less than or equal to  $x$ . For each  $k$  find the set  $A_{k}$  containing all integer values of the sequence  $(\sqrt[k]{a_{n}})_{n\geq 0}$ .

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