
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
1. The proposition "Some triangles are right-angled triangles" negated is "All triangles are not right-angled triangles".
2. If the inequality $ax^2-2x-1<0$ has solutions in the interval $[1, +\infty)$, then the range of the real number $a$ is $(-\infty, 3)$.
3. Given the function $f(x)=\sin(2x+\theta)$ ($\theta \in \mathbb{R}$), and for any $x \in \mathbb{R}$, $f\left( \frac{\pi}{2}-x \right)=-f(x)$, then $\sin(2\theta)=0$.
4. The function $f(x)=\cos x+ \frac{1}{\cos x}$ has a minimum value of 2 in the interval $\left(0, \frac{\pi}{2}\right)$.

Among these propositions, the correct ones are numbered as $\boxed{\text{1, 2, 3}}$.

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