
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. If $(a-2i)i=b-i$, where $a$, $b\in R$, $i$ is the imaginary unit, then $a^2+b^2=$______
2. In the geometric sequence $\{a_n\}$, if $a_9=1$, then $(a_1\cdot a_2\cdot \cdots \cdot a_n=a_1\cdot a_2\cdots a_{17-n}(n < 17))$ holds true, where $n\in\mathbf{N}^{*}$. Analogous to this property, in the arithmetic sequence $\{b_n\}$, if $b_7=0$, then ________.
3. Given the set $\{a,b,c\}=\{0,2,4\}$, and the following three relationships: $①a\neq 2$; $②b=2$; $③c\neq 0$. Only one of these relationships is correct, then $100a+10b+c$ equals ________.
4. In the plane rectangular coordinate system $xOy$, if the curve $y=ax^{2}+\frac{b}{x}(a,b$ are constants) passes through point $P(2,-5)$, and the tangent line of the curve at point $P$ is parallel to the line $7x+2y+3=0$, then the value of $a+b$ is ________.

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