
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
For any real numbers $a$ and $b$, the function $F(a,b)= \frac {1}{2}(a+b-|a-b|)$. If the function $f(x)=\sin x$, $g(x)=\cos x$, then for the function $G(x)=F(f(x),g(x))$. Consider the following five statements:  
$(1)$ The range of the function $G(x)$ is $[- \sqrt {2},2]$;  
$(2)$ $G(x) < 0$ if and only if $2k\pi+ \frac {\pi}{2} < x < 2(k+1)\pi(k\in\mathbb{Z})$;  
$(3)$ The function reaches its maximum value $1$ if and only if $x=2k\pi+ \frac {\pi}{2}(k\in\mathbb{Z})$;  
$(4)$ The distance between two adjacent highest points of the graph of $G(x)$ on $[ \frac {\pi}{4}, \frac {9\pi}{4}]$ is four times the distance between two adjacent lowest points;  
$(5)$ For any real number $x$, $G( \frac {5\pi}{4}-x)=G( \frac {5\pi}{4}+x)$ always holds.  
The correct statement numbers are __________.

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