
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
In the Cartesian coordinate system $xOy$, the parametric equations for line $C$ are
$$
\begin{cases}
x = 2 + t \\
y = t + 1
\end{cases}
(t \text{ is the parameter}),
$$
and the polar equation for curve $P$ in the polar coordinate system with $O$ as the pole and the positive x-axis as the polar axis is $\rho^2 - 4\rho\cos\theta + 3 = 0$.
(1) Find the standard equation of line $C$ and the Cartesian coordinate equation of curve $P$;
(2) Suppose the points of intersection between line $C$ and curve $P$ are $A$ and $B$, calculate the length of $|AB|$.

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