
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
(1) Given the coordinates of the three vertices of $\triangle ABC$ as $A(4,1)$, $B(0,3)$, and $C(2,4)$, and the midpoint of edge $AC$ as $D$, find the equation of the line containing the median $BD$ and express it in general form;
(2) Given that the center of circle $C$ is the intersection of the lines $2x+y+1=0$ and $x+3y-4=0$, and it is tangent to the line $3x+4y+17=0$, find the equation of circle $C$.

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