
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  $S_1, S_2, \dots, S_{2^n - 1}$  be the nonempty subsets of  $\{1, 2, \dots, n\}$  in some order, and let  $M$  be the  $(2^n - 1) \times (2^n - 1)$  matrix whose  $(i, j)$  entry is
\[m_{ij} = \left\{
\begin{array}{cl}
0 & \text{if  $S_i \cap S_j = \emptyset$ }, 
1 & \text{otherwise}.
\end{array}
\right.\]
Calculate the determinant of  $M$ .

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