
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 set with  $n$  elements, how many subsets are there whose cardinality is respectively  $\equiv 0$  (mod  $3$ ),  $\equiv 1$  (mod  $3$ ),   $ \equiv 2$  (mod  $3$ )? In other words, calculate  $$ s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k} $$  for  $i=0,1,2$ . Your result should be strong enough to permit direct evaluation of the numbers  $s_{i,n}$  and to show clearly the relationship of   $s_{0,n}, s_{1,n}$  and  $s_{2,n}$  to each other for all positive integers  $n$ . In particular, show the relationships among these three sums for  $n = 1000$ .

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