
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 parametric equation of line $l$: 
$$
\begin{cases}
x=1+t\cos \theta \\
y=t\sin \theta
\end{cases} 
(t \text{ is the parameter}),
$$
and the parametric equation of curve $C$: 
$$
\begin{cases}
x= \sqrt{3}\cos \alpha \\
y=\sin \alpha
\end{cases} 
(\alpha \text{ is the parameter}),
$$
and line intersects curve $C$ at points $A$ and $B$.

$(1)$ Convert the parametric equation of curve $C$ into a standard equation;

$(2)$ Given point $P(1,0)$, find the range of $|PA|\cdot|PB|$ as the inclination angle $\theta$ varies.

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