
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 $S$ denote the sum of all four to six-digit integers that have distinct digits. Also, each digit's contribution to the sum is its square multiplied by the positional value (1000 for the thousand's place, 100 for the hundred's place, 10 for the ten's place, and 1 for the units place). Compute the remainder when $S$ is divided by $1000$.

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