
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
The following propositions are given:<br/>① If "$a+b\neq 5$, then $a\neq 2$ or $b\neq 3$" is a true proposition;<br/>② The negation of the proposition "$\forall x\in R$, $x^{2}+x-2 \gt 0$" is "$\forall x\in R$, $x^{2}+x-2\leqslant 0$";<br/>③ For all $x \gt 0$, $x+\frac{1}{x}\geqslant a$ is a true proposition, then the maximum value of $a$ is $2$.<br/>The correct propositions are ______ (fill in the numbers).

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