
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 that $AB$ and $CD$ are two chords of the curve $C$: $\begin{cases} x=4t^{2} \\ y=4t \end{cases}$ (where $t$ is a parameter) intersecting at point $P(2,2)$, and if $AB \perpendicular CD$, and $|PA| \cdot |PB| = |PC| \cdot |PD|$.  
$(1)$ Convert the curve $C$ into a standard equation and explain what type of curve it is;  
$(2)$ Try to find the equation of line $AB$.

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