
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
For the function $y=f(x)$, if the function $F(x)=f(x+1)-f(x)$ is an increasing function, then the function $y=f(x)$ is said to have property $A$.

(1) If $f(x)=x^{2}+2$, find the analytical expression of $F(x)$ and determine whether $f(x)$ has property $A.

(2) Determine whether the proposition "A decreasing function does not have property $A$" is true, and explain the reason.

(3) If the function $f(x)=kx^{2}+x^{3}$ $(x\geqslant 0)$ has property $A$, find the range of real number $k$, and discuss the number of zeros of the function $g(x)=f(\sin x)-\sin x$ on the interval $[0,\pi]$.

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