
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
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a  $1$  on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins  $3$  games before Jacob wins  $3$  games can be written as  $\tfrac{p}{2^q}$ , where  $p$  and  $q$  are positive integers, and  $p$  is odd. Find the remainder when  $p+q$  is divided by  $1000$ .

*Proposed by firebolt360*

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