
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 \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \).

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