
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
Among the following statements:
1. The function $f(x)=\ln x+3x-6$ has only one zero and it belongs to the interval $(1,2)$;
2. If the inequality $ax^{2}+2ax+1 > 0$ with respect to $x$ always holds, then $a\in(0,1)$;
3. The graph of the function $y=x$ and the graph of the function $y=\sin x$ have 3 different intersection points;
4. The function $y=\sin x\cos x+\sin x+\cos x, x\in[0, \frac{\pi}{4}]$ has a minimum value of $1$.

The correct statements are __________. (Please write down the numbers of all the statements you think are correct.)

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