
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 rectangular coordinate system $xOy$, the parametric equations of line $C$ are given by $\begin{cases} x = 2 + t \ y = t + 1 \end{cases}$ ($t$ is a parameter), and the equation of curve $P$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis is given by $ρ^2 - 4ρ \cos θ + 3 = 0$.

   (a) Find the ordinary equation of line $C$ and the rectangular coordinate equation of curve $P$.

   (b) Let $A$ and $B$ be the intersection points of line $C$ and curve $P$. Find $|AB|$.

(2) Let $f(x) = |2x + 3| + |x - 1|$.

   (a) Solve the inequality $f(x) > 4$.

   (b) If there exists $x \in \left[-\frac{3}{2}, 1\right]$ such that the inequality $a + 1 > f(x)$ holds, find the range of values of the real number $a$.

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