
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 $\odot C$ that satisfies the following three conditions:
(1) $\odot C$ is tangent to the $x$-axis;
(2) The center $C$ is on the line $3x - y = 0$;
(3) $\odot C$ intersects the line $x - y = 0$ at points $A$ and $B$, and the area of $\triangle ABC$ is $\sqrt{14}$. 

Find the equation of the circle $\odot C$ that meets the above conditions.

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