
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 curves $C_1:\begin{cases} & x=-2+\cos t \\ & y=1+\sin t \end{cases}$ ($t$ is the parameter), $C_2:\begin{cases} & x=4\cos \theta \\ & y=3\sin \theta \end{cases}$ ($\theta$ is the parameter).

(I) Convert $C_1$ and $C_2$ into standard equations and explain what curves they represent;

(II) A line $l$ that passes through the left vertex of curve $C_2$ and has a slope angle of $\frac{\pi}{4}$ intersects curve $C_1$ at points $A$ and $B$. Calculate $|AB|$.

(2) Given the function $f(x)=|a-3x|-|2+x|$.

(I) If $a=2$, solve the inequality $f(x)\leqslant 3$;

(II) If there exists a real number $x$ such that the inequality $f(x)\geqslant 1-a+2|2+x|$ holds, find the range of the real number $a$.

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