
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
The Vlček family was pressing apple juice. They had two equally sized barrels, each containing almost the same amount of juice. If they poured 1 liter from the first barrel into the second barrel, they would have the same amount in both barrels, but neither barrel would be full. So instead, they poured 9 liters from the second barrel into the first. Then, the first barrel was completely full, and the juice in the second barrel filled exactly one-third of its volume. How many liters of juice did they press, what was the volume of the barrels, and how much juice was originally in each barrel?

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