
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
Let  $n$  be a positive integer. David has six  $n\times n$  chessboards which he arranges in an  $n\times n\times n$  cube. Two cells are "aligned" if they can be connected by a path of cells  $a=c_1,\ c_2,\ \dots,\ c_m=b$  such that all consecutive cells in the path share a side, and the sides that the cell  $c_i$  shares with its neighbors are on opposite sides of the square for  $i=2,\ 3,\ \dots\ m-1$ . 

Two towers attack each other if the cells they occupy are aligned. What is the maximum amount of towers he can place on the board such that no two towers attack each other?

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