
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
For a positive integer  $n$ , define  $d(n)$  to be the sum of the digits of  $n$  when written in binary (for example,  $d(13)=1+1+0+1=3$ ). Let
\[
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
\]
Determine  $S$  modulo  $2020$ .

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