
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
Consider a grid with  $n{}$  lines and  $m{}$  columns  $(n,m\in\mathbb{N},m,n\ge2)$  made of  $n\cdot m \; 1\times1$  squares called  ${cells}$ . A  ${snake}$  is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the  ${length}$  of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.   

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