
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 two lines $l_1: y=2x+3$ and $l_2: y=x+2$ intersect at point C.  
(1) Find the coordinates of point C;  
(2) Find the equation of the circle with center C and tangent to the line $3x+4y+4=0$;  
(3) If the line $x+y+t=0$ intersects the circle C at points A and B, find the maximum value of the area of $\triangle ABC$ and the value of the real number $t$.

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