
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 the set \( A = \{-100, -50, -1, 1, 2, 4, 8, 16, 32, 2003\} \). All non-empty subsets of $A$ are denoted as $A_i$, for $i = 1, 2, \cdots, 1023$. The product of all elements in each subset $A_i$ is denoted as $a_i$, for $i = 1, 2, \cdots, 1023$. Calculate the value of $a_1 + a_2 + \cdots + a_{1023}$.

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