
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) Let vectors $\overrightarrow{a}=(m,1), \overrightarrow{b}=(1,2)$, and it is given that ${{\left| \overrightarrow{a}+\overrightarrow{b} \right|}^{2}}={{\left| \overrightarrow{a} \right|}^{2}}+{{\left| \overrightarrow{b} \right|}^{2}}$, then $m=$ ______.

(2) In the expansion of ${{(2x+\sqrt{x})}^{5}}$, the coefficient of ${{x}^{3}}$ is ______. (Fill in the answer with numbers)

(3) Given that $f(x)$ is an even function defined on $\mathbb{R}$, and is monotonically increasing on the interval $(-\infty ,0)$. If the real number $a$ satisfies $f\left({2}^{\left|a-1\right|}\right) > f\left(- \sqrt{2}\right)$, then the range of values for $a$ is ______.

(4) Given an ellipse $\Gamma : \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$ with its right focus at $F(3,0)$, and its top and bottom vertices as $A,B$ respectively. The line $AF$ intersects $\Gamma$ at another point $M$, and the line $BM$ intersects the $x$-axis at point $N(12,0)$. The eccentricity of $\Gamma$ is ______.

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