
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 minimum value of the quadratic function $f(x)=x^{2}-x$ is $-\dfrac{1}{4}$. Among the following inequalities:

$(1) m-\sqrt{m}\geqslant -\dfrac{1}{4}$,

$(2) m+n+\dfrac{1}{2} < \sqrt{m}+\sqrt{n}$,

$(3) \dfrac{1}{2}(m+n)^{2}+\dfrac{1}{4}(m+n)\geqslant m\sqrt{n}+n\sqrt{m}$

which ones always hold true over $[0,+\infty)$? The sequence number is ______.

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