
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
A. In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x = 2 + t \cos \alpha \\
y = 1 + t \sin \alpha
\end{cases}
$$
where $t$ is a parameter. In the polar coordinate system, with the same unit length as $xOy$, the origin $O$ as the pole, and the positive x-axis as the polar axis, the equation of circle $C$ is 
$$
\rho = 6 \cos \theta.
$$

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

(2) Let circle $C$ and line $l$ intersect at points $A$ and $B$. If point $P$ has coordinates $(2,1)$, find the minimum value of $|PA| + |PB|$.

B. Let the function be $f(x) = |x + 2| - |x - 2|$.

(1) Find the solution set of the inequality $f(x) > 1$.

(2) If the inequality $f(x) + 4 \geqslant |1 - 2m|$ has a solution for $x$, find the range of real numbers for $m$.

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