
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
Ryan is learning number theory. He reads about the *Möbius function*  $\mu : \mathbb N \to \mathbb Z$ , defined by  $\mu(1)=1$  and
\[ \mu(n) = -\sum_{\substack{d\mid n  d \neq n}} \mu(d) \]
for  $n>1$  (here  $\mathbb N$  is the set of positive integers).
However, Ryan doesn't like negative numbers, so he invents his own function: the *dubious function*  $\delta : \mathbb N \to \mathbb N$ , defined by the relations  $\delta(1)=1$  and
\[ \delta(n) = \sum_{\substack{d\mid n  d \neq n}} \delta(d) \]
for  $n > 1$ . Help Ryan determine the value of  $1000p+q$ , where  $p,q$  are relatively prime positive integers satisfying
\[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}.  \]

*Proposed by Michael Kural*

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