
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
This examination consists of three optional questions (1), (2), and (3), each worth 7 points. Candidates should select any 2 questions to answer for a total of 14 points. If more than two questions are completed, only the first two answered will be scored. When answering, first darken the corresponding question number on the answer card with a 2B pencil, and enter the selected question number in the brackets.
(1) Elective 4-2: Matrices and Transformations
Find the eigenvalues and the corresponding eigenvectors of the matrix $$A= \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix}$$.
(2) Elective 4-4: Coordinate Systems and Parametric Equations
Given the parametric equations of the line l: $$\begin{cases} x=t \\ y=1+2t \end{cases}$$ (where t is the parameter) and the polar equation of the circle C: $$\rho =2\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$$.
(Ⅰ) Convert the parametric equations of line l into a standard equation and the polar equation of circle C into a rectangular coordinate equation.
(Ⅱ) Determine the positional relationship between line l and circle C.
(3) Elective 4-5: Inequality Elective Topic
Given the function \(f(x)=|x-1|+|x-2|\). If the inequality \(|a+b|+|a-b|\geq |a|f(x)\) (with \(a \neq 0\), \(a, b \in R\)) is always satisfied, determine the range of the real number \(x\).

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