
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 that the pole $O$ of the polar coordinate system coincides with the origin of the rectangular coordinate system, and the polar axis coincides with the positive semi-axis of the $x$-axis in the rectangular coordinate system. The parametric equations of the circle $C$ are $\begin{cases} x=a+a\cos θ \\ y=a\sin θ \end{cases}$ ($θ$ is the parameter, $0 < a < 5$). The polar equation of the line $l$ is $ρ\sin \left(θ+ \dfrac{π}{4} \right) =2\sqrt{2}$. If the line $l$ intersects the circle $C$ at points $A$ and $B$, and $|AB|=2\sqrt{2}$.

1. Find the value of $a$.
2. If $M$ and $N$ are two distinct points on the circle $C$, and $\angle MON=\dfrac{\pi }{3}$, find the maximum value of $|OM|+|ON|$.

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