
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 equation $x^{2}+y^{2}-2(t+3)x+2(1-4t^{2})y+16t^{4}+9=0$, which represents a circle. Find:

(1) The range of $t$.
(2) The value of $t$ when the radius $r$ is at its maximum.

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