
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
The expression  $16^n+4^n+1$  is equiavalent to the expression  $(2^{p(n)}-1)/(2^{q(n)}-1)$  for all positive integers  $n>1$  where  $p(n)$  and  $q(n)$  are functions and  $\tfrac{p(n)}{q(n)}$  is constant.  Find  $p(2006)-q(2006)$ .

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