
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: $x^2+y^2-2(t+3)x+2(1-4t^2)y+16t^4+9=0$ ($t \in \mathbb{R}$) describes the trajectory of a point on a circle.  
(1) Find the range of values for $t$;  
(2) Find the equation of the circle with the largest area;  
(3) If point $P(3, 4t^2)$ is always inside the given circle, find the range of values for $t$.

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