
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
Find the number of sets  $A$  containing  $9$  positive integers with the following property: for any positive integer  $n\le 500$ , there exists a subset  $B\subset A$  such that  $\sum_{b\in B}{b}=n$ .

*Bogdan Enescu & Dan Ismailescu*

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