
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) For the complex number $z=\cos 75^\circ +i\sin 75^\circ$ (where $i$ is the imaginary unit), the point corresponding to $z^2$ in the complex plane is located in the \_\_\_\_\_ quadrant.

(2) Given $x+2y=1$, the minimum value of $2^x+4^y$ is \_\_\_\_\_.

(3) Given points $A(x_1,x_1^2)$ and $B(x_2,x_2^2)$ are any two distinct points on the parabola, according to the graph, the line segment $AB$ is always above the graph of the function between points $A$ and $B$, thus the conclusion $\frac{x_1^2+x_2^2}{2} > \left(\frac{x_1+x_2}{2}\right)^2$ holds. By analogy, if points $A(x_1,\sin x_1)$ and $B(x_2,\sin x_2)$ are two distinct points on the graph of the function $y=\sin x$ ($x\in(0,\pi)$), then similarly, the conclusion is \_\_\_\_\_\_.

(4) Given $f(x)$ is a quadratic function, and the solution set of the inequality $f(x) < 0$ is $(-2017,2019)$. If $f(t-1) > f(1+t^2)$, then the range of real numbers $t$ is \_\_\_\_\_\_.

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