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

(Ⅰ) Convert the parametric equation of curve $C$ into a standard form and find the length of $|AB|$ when $\theta = \frac{\pi}{4}$.

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

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