
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 \( S = \{1,2,3, \cdots, 10\} \). Given that subset \( A \) of \( S \) satisfies \( A \cap \{1,2,3\} \neq \varnothing \) and \( A \cup \{4,5,6\} \neq S \), determine the number of such subsets \( A \). 

(Note: The original problem includes determining the count of subset \( A \) that meets the given conditions.)

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