
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 eight different symbols designed on  $n\geq 2$  different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any  $k$  symbols  $(1\leq k\leq 7)$  the number of shirts containing at least one of the  $k$  symbols is even. Determine the value of  $n$ .

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