
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 $$\begin{cases} x=1+ \frac { \sqrt {3}}{2}t \\ y=a-1+ \frac {1}{2}t\end{cases}$$ (where parameter $t\in\mathbb{R}$, $a$ is a constant), in the polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis, the equation of curve $C$ is $\rho=2\sqrt{2}\cos\left(\theta+ \frac{\pi}{4}\right)$.  
(1) Find the standard equation of curve $C$;  
(2) Given that line $l$ intersects curve $C$ at points $A$ and $B$ and $|AB|= \sqrt{5}$, find the value of the constant $a$.

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