
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 polar coordinate system, the equation of circle $C$ is $\rho=2a\sin \theta (a\neq 0)$, with the pole as the origin of the coordinate system and the polar axis as the positive half-axis of the $x$-axis, a Cartesian coordinate system is established. The parametric equation of the line is given by $\begin{cases} x=t+2 \\ y=2t+3\end{cases}(\theta$ is the parameter).

$(1)$ Find the Cartesian coordinate equation of circle $C$ and the standard equation of the line;  
$(2)$ If line $l$ always has common points with circle $C$, find the range of the real number $a$.

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