
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 the definition of the characteristic function $f\_P(x)$ of a non-empty subset $P$ of the universal set $U$ as follows:

$f\_P(x) = \begin{cases} 1, & x \in P \ 0, & x \in \complement\_U P \end{cases}$

where $\complement\_U P$ denotes the complement of set $P$ in the universal set $U$. Suppose $A$ and $B$ are both non-empty subsets of $U$. The following propositions are given:

1. If $A \subseteq B$, then for any $x \in U$, $f\_A(x) \leq f\_B(x)$.
2. For any $x \in U$, $f\_{\complement\_U A}(x) = 1 - f\_A(x)$.
3. For any $x \in U$, $f\_{A \cap B}(x) = f\_A(x) \cdot f\_B(x)$.
4. For any $x \in U$, $f\_{A \cup B}(x) = f\_A(x) + f\_B(x)$.

Determine the sequence number(s) of the correct proposition(s).

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