
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
Given a positive integer $n$. For a family of subsets $\mathscr{D}$ of the set $\{1,2, \cdots, n\}$, suppose it satisfies the following condition: if $A$ is a member of the family of subsets $\mathscr{D}$ and $B$ is a subset of $A$, then $B$ is also a member of the family of subsets $\mathscr{D}$. Such a family of subsets $\mathscr{D}$ is called "downward closed".

For a downward closed family of subsets $\mathscr{D}$, find the maximum value of the expression $\sum_{A \in \mathscr{Q}}(-1)^{|A|}$, where $|A|$ denotes the size of the subset $A$.

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