
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 $xoy$, a polar coordinate system is established with $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the line $l$ is $\theta=\frac{\pi}{4}\left(\rho\in\mathbb{R}\right)$, and the parametric equation of the curve $C$ is $\begin{cases} x=\sqrt{2}\cos\theta \\ y=\sin\theta \end{cases}\left(\theta\text{ is the parameter}\right)$.

1. Write the cartesian coordinate equations of line $l$ and curve $C$.
2. The line passing through point $M$ and parallel to line $l$ intersects with curve $C$ at points $A$ and $B$. If $|MA|\cdot|MB|=\frac{8}{3}$, find the cartesian coordinate equation of the trajectory of point $M$, and describe the shape of the trajectory.

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