
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
If $X$ is a non-empty set and $M$ is a set consisting of some subsets of $X$, and it satisfies:
1. $X \in M$, $\emptyset \in M$;
2. For any subsets $A, B$ of $X$, if $A \in M$ and $B \in M$, then $A \cup B \in M$;
3. For any subsets $A, B$ of $X$, if $A \in M$ and $B \in M$, then $A \cap B \in M$;

Then $M$ is called an "M-set class" of set $X$.
For example: $M=\{\emptyset, \{b\}, \{c\}, \{b, c\}, \{a, b, c\}\}$ is an "M-set class" of set $X=\{a, b, c\}$. Given the set $X=\{a, b, c\}$, the number of "M-set classes" containing $\{b, c\}$ is $\boxed{10}$.

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