
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 definite integral $\int_{-1}^{1}(x^{2}+\sin x)dx=$ ______.

(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.

(3) Given the function $f(x)=\begin{cases}1-\log_{a}(x+2), & x\geqslant 0 \\ g(x), & x < 0\end{cases}$ is an odd function, then the root of the equation $g(x)=2$ is ______.

(4) Given the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(0 < b < a < \sqrt{2}b)$, its foci are $F_{1}$ and $F_{2}$ respectively. Circle $N$ has $F_{2}$ as its center, and its minor axis length as the diameter. A tangent line to circle $N$ passing through point $F_{1}$ touches it at points $A$ and $B$. If the area of quadrilateral $F_{1}AF_{2}B$ is $S= \frac{2}{3}a^{2}$, then the eccentricity of ellipse $M$ is ______.

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