
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 a circle $C$: $x^{2}+y^{2}-2ax+(4-2a)y+9a+3=0$, the distance from the center of circle $C$ to the line $l$: $3x+4y+8=0$ is $\frac{7}{5}$.

(1) Find the value of $a$;

(2) If point $C$ is symmetric to the origin point $O$, and point $A$ is a moving point on circle $C$. On line segment $AD$, take a point $B$ that satisfies $\frac{|DB|}{|BA|}=3$, find the trajectory equation of point $B$.

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