
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
Given that Ace travels at a constant speed, and when traveling east, Flash runs \( x \) times as fast as Ace, and when traveling west, Flash runs \( x+1 \) times as fast as Ace, with \( x>1 \), determine the total distance Flash must run to first catch Ace traveling east and then catch Ace again after turning around to travel west, where Flash has a head start of \( 2y \) yards when traveling east, and a head start of \( y \) yards when traveling west.

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