
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
Compute the sum:
\[
\sum_{1 \le a < b < c < d} \frac{1}{2^a 4^b 8^c 16^d}.
\]
(The sum is taken over all quadruples $(a,b,c,d)$ of positive integers such that $1 \le a < b < c < d.$)

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