
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
Barbara and Jenna play a modified game, where they take turns to remove coins from a table. On her turn, Barbara can remove $3$ or $5$ coins, while Jenna can remove $1$ or $3$ coins. Whoever removes the last coin wins. A coin toss decides who starts first. Determine the winner when the game starts with $1001$ coins and when it starts with $1002$ coins using optimal strategies.
A) Barbara wins with 1001 coins, Jenna wins with 1002 coins.
B) Barbara wins with both 1001 and 1002 coins.
C) Jenna wins with 1001 coins, Barbara wins with 1002 coins.
D) Jenna wins with 1001 coins, and the starter in 1002 wins.
E) The starter in 1001 loses, and the starter in 1002 wins.

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