
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]


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Given the polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. Point $A(4, \frac{5\pi}{4})$ is known. The polar coordinate equation of curve $E$ is $ρ=ρcos^2θ+\sqrt{2}acosθ (a > 0)$. A perpendicular line $l$ is drawn through point $A$ intersecting curve $E$ at points $B$ and $C$ when $θ=\frac{3\pi}{4} (\rho \in R)$.  
$(1)$ Write down the rectangular coordinate equations of curve $E$ and line $l$.  
$(2)$ If $|AB|$, $|BC|$, and $|AC|$ form a geometric progression, find the value of the real number $a$.

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