
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 the function $f(x)=\begin{cases} x+1,x\in [-1,0) \\\\ \sqrt{1-{{x}^{2}}},x\in [0,1] \\end{cases}$, find the value of the definite integral $\int_{-1}^{1}f(x)dx$;

(2) If complex numbers ${{z}\_{1}}=a+2i\left( a\in R \right)$ and ${{z}\_{2}}=3-4i$, and $\frac{{{z}\_{1}}}{{{z}\_{2}}}$ is a purely imaginary number, find $\left| {{z}\_{1}} \right|$.

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