
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
There are several propositions:
$(1)$ The distance between adjacent symmetry centers in the graph of the function $y=\cos (x-\frac{π}{4})\cos (x+\frac{π}{4})$ is $π$;
$(2)$ The negation of the proposition "If $a=0$, then $ab=0$" is "If $a=0$, then $ab\neq 0$";
$(3)$ "$a\neq 5$ and $b\neq -5$" is a necessary but not sufficient condition for "$a+b\neq 0$";
$(4)$ Given the proposition $p$: For any $x\in R$, $\sin x\leqslant 1$, then $\lnot p$ is: There exists $x_{0}\in R$ such that $\sin x_{0} > 1$;
$(5)$ The proposition "If $0 < a < 1$, then $\log_{a}(a+1) > \log_{a}(1+ \frac{1}{a})$" is true;
$(6)$ The inequality $| \vec{a}- \vec{b}|\leqslant | \vec{a}+ \vec{b}|$ always holds;
$(7)$ If $\vec{a}\cdot \vec{b}=0$, then $\vec{a}\perp \vec{b}$;
The sequence numbers of all true propositions are $\_\_\_\_\_\_$.

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