
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 $X$ is a non-empty set and $M$ is a set containing some subsets of $X$, and if:
(1) $X \in M$, $\Phi \in M$;
(2) For any subsets $A$ and $B$ of $X$, if $A \in M$ and $B \in M$, then $A \cup B \in M$ and $A \cap B \in M$.
Then $M$ is called an "M-set class" of the set $X$.
For example: $M=\{\Phi, \{b\}, \{c\}, \{b, c\}, \{a, b, c\}\}$ is an "M-set class" of the set $X=\{a, b, c\}$. Given the set $X=\{a, b, c\}$, the number of "M-set classes" that contain $\{b, c\}$ is _______.

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