
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 equation of curve $C_1$ is 
\[
\begin{cases}
x=4t^2 \\
y=4t
\end{cases}
\]
where $t$ is the parameter. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit length. The polar equation of curve $C_2$ is $p\cos \left(\theta+ \frac{\pi}{4}\right)= \frac{ \sqrt{2}}{2}$.

(Ⅰ) Convert the equation of curve $C_1$ into a standard equation, and the equation of $C_1$ into a Cartesian coordinate equation;

(Ⅱ) If curves $C_1$ and $C_2$ intersect at points $A$ and $B$, and the midpoint of $AB$ is $P$, a perpendicular line to curve $C_1$ at point $P$ intersects curve $C_1$ at points $E$ and $F$, find $|PE|\cdot|PF|$.

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