
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 plane rectangular coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar coordinate equation of curve $C$ is $\rho =2\sin \theta +2a\cos \theta \left(a \gt 0\right)$; the parametric equation of line $l$ is $\left\{\begin{array}{l}x=-2+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Line $l$ intersects curve $C$ at points $M$ and $N$ respectively.<br/>$(1)$ Write down the rectangular coordinate equation of curve $C$ and the standard equation of line $l$;<br/>$(2)$ If the polar coordinates of point $P$ are $\left(2,\pi \right)$, and $|{PM}|+|{PN}|=5\sqrt{2}$, find the value of $a$.

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