
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. In the complex plane, the distance from the point corresponding to the complex number $z=-2i+1$ to the origin is $\_\_\_\_\_\_\_\_$.
2. Given ${{2}^{a}}={{5}^{b}}=\sqrt{10}$, find the value of $\frac{1}{a}+\frac{1}{b}=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$.
3. Suppose the function $f(x)=g(x)+x^{2}$, and the tangent line equation of the curve $y=g(x)$ at the point $(1,g(1))$ is $9x+y-1=0$. Find the tangent line equation of the curve $y=f(x)$ at the point $(1,f(1))$ is $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$
4. Given the function $f(x)=\sin ^{2}x+a\cos x+a$, $a\in R$. If for any $x$ in the interval $[0,\frac{\pi }{2}]$, $f(x)\leqslant 1$ holds, find the range of values for $a$ is $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

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