
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 the circle $(x^{2}+y^{2}+(4-2a)x-2\sqrt{3}ay+4a^{2}-4a-12=0)$, determine the line $l$ passing through point $A(1,0)$. If, for any real number $a$, the length of the chord intercepted by line $l$ on the circle $C$ is always a fixed value $d$, find the value of $d$.

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