
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  $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$ . In other words,  $a_n$  is the sum of  $\frac{1}{2^{d+ \frac{n}{d}}}$  over all divisors  $d$  of  $n$ . 
Find  $$ \frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....} $$  

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