
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) Evaluate the integral: $\int_{0}^{1}(\sqrt{1-x^{2}}+x+x^{3})dx$

(2) Evaluate: $\frac{\cos 20^{\circ}}{\cos 35^{\circ} \sqrt{1-\sin 20^{\circ}}}$

(3) Determine the correct statements regarding lines $m$, $n$, $p$ and planes $α$, $β$, $γ$. (Write down the serial numbers of all correct statements.)
   (i) If $m \perp p$ and $m \parallel n$, then $n \perp p$;
   (ii) If $m \parallel β$, $n \parallel β$, $m \subset α$, and $n \subset α$, then $α \parallel β$;
   (iii) If $α \perp γ$, $β \perp γ$, and $α \cap β = m$, then $m \perp γ$;
   (iv) If $α \parallel β$, $m \subset α$, and $n \subset β$, then $m \parallel n$.

(4) Let the domain of the function $y=f(x)$ be $D$. If for any $x_1, x_2 \in D$, when $x_1 + x_2 = 2a$, it is always true that $f(x_1) + f(x_2) = 2b$, then the point $(a, b)$ is called the center of symmetry of the graph of $y=f(x)$. Investigate a symmetry point of the graph of the function $f(x)=x^3+\sin x+2$, and use the definition of the center of symmetry to determine the value of $f(-1)+f(-\frac{9}{10})+\ldots+f(0)+\ldots+f(\frac{9}{10})+f(1)$.

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