
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) **Coordinate System and Parametric Equations**

In the polar coordinate system, it is known that the circle $C$ has its center at $C(\sqrt{2},\frac{\pi}{4})$ and radius $r=\sqrt{3}$.

(1) Find the Cartesian coordinate equation of circle $C$;

(2) If $\alpha \in [0,\frac{\pi}{4})$, the parametric equation of line $l$ is $\begin{cases} & x=2+t\cos \alpha \\ & y=2+t\sin \alpha \\ \end{cases}$ (where $t$ is the parameter), and line $l$ intersects circle $C$ at points $A$ and $B$. Find the range of values for the chord length $|AB|$.

(2) **Inequalities**

Given $f(x)=|x+1|+|x-1|$.

(1) Find the solution set of the inequality $f(x) < 4$;

(2) If the inequality $f(x)-|a+1| < 0$ has a solution, find the range of values for $a$.

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