
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 function $y=f(x)$ has a graph on which two different points $A(x_1,y_1)$ and $B(x_2,y_2)$ have slopes of their tangents as $k_A$ and $k_B$ respectively. Define $\varphi(A,B)= \frac{|k_A-k_B|}{|AB|}$ (where $|AB|$ is the length of segment $AB$) as the "curvature" of the curve $y=f(x)$ between points $A$ and $B$. Consider the following propositions:

1. For the function $y=x^3-x^2+1$, with points $A$ and $B$ having $x$-coordinates of $1$ and $2$ respectively, then $\varphi(A,B) > \sqrt{3}$;
2. There exists such a function whose graph has a constant "curvature" between any two points;
3. Let points $A$ and $B$ be two different points on the parabola $y=x^2+1$, then $\varphi(A,B)\leqslant 2$;
4. Let the curve be $y=e^x$ (where $e$ is the base of the natural logarithm) with two different points $A(x_1,y_1)$ and $B(x_2,y_2)$, and $x_1-x_2=1$. If $t\cdot\varphi(A,B) < 1$ always holds, then the range of real numbers for $t$ is $(-\infty,1)$. Among these propositions, the numbers of the true ones are ______.

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