
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
The function \( f(x) = \begin{cases} 
x & \text{if } x \in P \\
-x & \text{if } x \in M 
\end{cases} \)
where \( P \) and \( M \) are two non-empty subsets of the real numbers \(\mathbb{R}\). Additionally, it is given that \( f(P) = \{ y \mid y = f(x), x \in P \} \) and \( f(M) = \{ y \mid y = f(x), x \in M \} \). Consider the following statements:
1. If \( P \cap M = \varnothing \), then \( f(P) \cap f(M) = \varnothing \).
2. If \( P \cap M \neq \varnothing \), then \( f(P) \cap f(M) \neq \varnothing \).
3. If \( P \cup M = \mathbb{R} \), then \( f(P) \cup f(M) = \mathbb{R} \).
4. If \( P \cup M \neq \mathbb{R} \), then \( f(P) \cup f(M) \neq \mathbb{R} \).

Which of the statements are correct?

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