
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 function $f(x) = \cos x + x$, where $x \in \mathbb{R}$, and $\sin x_0 = 0$, $x_0 \in \mathbb{R}$, then among the following propositions, the true ones are \_\_\_\_\_\_\_\_  
① The maximum value of $f(x)$ is $f(x_0)$; ② The minimum value of $f(x)$ is $f(x_0)$;  
③ $f(x)$ is an increasing function on $\mathbb{R}$; ④ $f(x)$ is an increasing function on $\mathbb{R}$

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