
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
A positive interger number  $k$  is called “ $t-m$ ”-property if forall positive interger number  $a$ , there exists a positive integer number  $n$  such that ${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$ a) Find all positive integer numbers  $k$  which has  $t-20$ -property.
b) Find smallest positive integer number  $k$  which has  $t-{{20}^{15}}$ -property.


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