
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 the set \( M = \{1, 2, \cdots, 2023\} \). For a subset \( A = \{a_1, \cdots, a_k\} \) of \( M \), define a corresponding number \( \alpha = (-1)^{a_1 + \cdots + a_k} \) (for the empty set \( \varnothing \), the corresponding number \( \alpha = (-1)^0 = 1 \)). Find the total sum of the corresponding numbers for all subsets \( A \) of \( M \), i.e., find \( \sum_{A \subseteq M} \alpha \).

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