
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
There are  $7$  balls in a jar, numbered from  $1$  to  $7$ , inclusive. First, Richard takes  $a$  balls from the jar at once, where  $a$  is an integer between  $1$  and  $6$ , inclusive. Next, Janelle takes  $b$  of the remaining balls from the jar at once, where  $b$  is an integer between  $1$  and the number of balls left, inclusive. Finally, Tai takes all of the remaining balls from the jar at once, if any are left. Find the remainder when the number of possible ways for this to occur is divided by  $1000$ , if it matters who gets which ball.

*Proposed by firebolt360 & DeToasty3*

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