
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
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The game rules state that the winner is the first to knock the bottle off the ledge. If the probability that Larry knocks off the bottle is $\tfrac{2}{3}$ each turn for the first four turns and $\tfrac{1}{2}$ for turns five and onwards, and Julius has a probability of $\tfrac{1}{3}$ each turn for the first four turns and $\tfrac{1}{2}$ for turns five and onwards, determine the probability that Larry wins the game.

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