
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 $f(x)=\sin x- \frac{1}{3}x,x\in[0,\pi]$. If $\cos x_{0}= \frac{1}{3}(x_{0}\in[0,\pi])$, then the sequence number of the true proposition(s) below is $\_\_\_\_\_\_$.

$①$ The maximum value of $f(x)$ is $f(x_{0})$

$②$ The minimum value of $f(x)$ is $f(x_{0})$

$③$ The function $f(x)$ is decreasing on $[0,x_{0}]$

$④$ The function $f(x)$ is decreasing on $[x_{0},\pi]$

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