
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
There are \( n \) pieces on a board with \( 5 \times 9 \) squares, with a maximum of one piece on each square at any time. A move consists of moving each piece to one of the adjacent squares at the top, bottom, left, or right. This is done simultaneously for all pieces. If a piece is moved horizontally in one turn, it must be moved vertically in the next turn and vice versa. Determine the largest value for \( n \) so that there is a starting position for the \( n \) pieces and a sequence of moves so that this game can continue for as long as you like.

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