
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 pair of integer numbers  $(n,k)$  such that  $n$  is not negative and  $k$  is greater than  $1$ , and satisfying that the number:
\[ A=17^{2006n}+4.17^{2n}+7.19^{5n}  \]
can be represented as the product of  $k$  consecutive positive integers.

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