
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
Given the curve $C$ with the polar coordinate equation $ρ=2\cos θ$, establish a rectangular coordinate system with the pole as the origin and the positive semi-axis of the polar axis as the $x$-axis. The parameter equation of line $L$ is $\begin{cases} x=\frac{\sqrt{3}}{2}t+m \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter).

(1) Find the rectangular coordinate equation of curve $C$ and the general equation of line $L$.
(2) Let point $P(m,0)$. If line $L$ intersects curve $C$ at points $A$ and $B$, and $|PA| \cdot |PB| = 1$, find the value of the real number $m$.

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