
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 propositions:  
(1) The diagonals of a trapezoid are equal;  
(2) Some real numbers are non-repeating infinite decimals;  
(3) There exists a real number $x$ such that $x^{2}+2x+3=0$;  
(4) $x^{2}\neq y^{2} \Leftrightarrow x\neq y$ or $x\neq -y$;  
(5) The contrapositive of the proposition "If $a$ and $b$ are both even, then $a+b$ is even" is "If $a+b$ is not even, then $a$ and $b$ are both not even";  
(6) If "either $p$ or $q$" is a false proposition, then "neither $p$ nor $q$" is a true proposition;  
(7) Given that $a$, $b$, and $c$ are real numbers, if the solution set of the inequality $ax^{2}+bx+c\leqslant 0$ is empty, then it must be that $a > 0$ and $\triangle \leqslant 0$.  
The numbers of the true propositions are __________.

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