
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
Vincent has two jars, one red and one blue, and a pile of 20 pebbles. Initially, both jars are empty. A move in Vincent's game consists of transferring a pebble from the pile to one of the jars or returning a pebble from one of the jars to the pile. The number of pebbles in the jars determines the game's position. After each move, the number of pebbles in the red jar is always at least as many as the number of pebbles in the blue jar, and during the game, no position can be repeated. What is the maximum number of moves Vincent can make?

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