
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 $C:x^{2}+y^{2}-2ax-2by+a^{2}+b^{2}-1=0$ $(a < 0)$ whose center is on the line $\sqrt{3}x-y+\sqrt{3}=0$, and the maximum distance from a point on circle $C$ to the line $\sqrt{3}x+y=0$ is $1+\sqrt{3}$, calculate the value of $a^{2}+b^{2}$.

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