
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
In the Cartesian coordinate system $xOy$, a circle $C_1$ is defined by the equation $x^2+y^2+2mx-(4m+6)y-4=0$ ($m\in\mathbb{R}$), and it intersects with another circle $C_2$ centered at $(-2, 3)$ at two points $A(x_1, y_1)$ and $B(x_2, y_2)$. It is also known that $x_1^2-x_2^2=y_2^2-y_1^2$. Determine the value of the real number $m$.

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