
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) Given $a=\int_{0}^{1}{(e^{x}+2x)dx}$ (where $e$ is the base of the natural logarithm), then the real number $a=$ ______.

(2) Given unit vectors $e_{1}$ and $e_{2}$ with an angle $\alpha$ between them, where $\cos \alpha =\frac{1}{3}$, and $a=3e_{1}-2e_{2}$, then $|a|=$ ______.

(3) Given $f(x)=A\cos ^{2}(\omega x+\varphi)+1$ ($A > 0$, $\omega > 0$, $0 < \varphi < \frac{\pi }{2}$) has a maximum value of $3$, the graph of $f(x)$ intersects the $y$-axis at the point $(0,2)$, and the distance between two adjacent centers of symmetry is $2$, then $f(1)+f(2)+\ldots+f(14)=$ ______.

(4) In the Cartesian coordinate system $xOy$, point $P$ is a common point of the graph of the function $y=2\ln x$ and the circle $M$: $(x-3)^{2}+y^{2}=r^{2}$, and they have a common tangent line at point $P$. If the graph of the quadratic function $y=f(x)$ passes through points $O$, $P$, and $M$, then the maximum value of the function $y=f(x)$ is ______.

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