
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 conclusions:
1. If $x > 0$, $y > 0$, then a sufficient but not necessary condition for "$x+2y=2 \sqrt {2xy}$" to hold is "$x=2$ and $y=1$";
2. There exists $a > 1$, $x > 0$, such that $a^{x} < \log_{a}x$;
3. If $f(x)$ is continuous on $[a,b)$ and $\int_{a}^{b}f(x)dx > 0$, then $f(x)$ is always positive on $[a,b)$;
4. In an acute-angled $\triangle ABC$, if $\sin B(1+2\cos C)=2\sin A\cos C+\cos A\sin C$, then it must be that $A=2B$;
5. The trajectory equation of a moving point $P$ on the plane, which is at a distance greater than $1$ from a fixed point $F(1,0)$ compared to its distance from the $y$-axis, is $y^{2}=4x$.

The correct conclusion numbers are __________.

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