
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 line $l$ is 
$$
\begin{cases} 
x=a+ \frac {3}{5}t \\
y=1+ \frac {4}{5}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation of curve $C$ is $\rho\cos ^{2}\theta+8\cos \theta-\rho=0$

$(I)$ Find the general equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
$(II)$ Given point $P(a,1)$, suppose line $l$ intersects curve $C$ at two points $A$ and $B$. If $|PA|=3|PB|$, find the value of $a$.

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