
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 polar coordinate system, the equation of the circle $C_1$ is given by $\rho=-2\sqrt{2}\cos\left(\theta-\frac{\pi}{4}\right)$. Establishing a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis, the parametric equation of the circle $C_2$ is 
$$\begin{cases}
x = 2 + m\cos\theta \\
y = 2 + m\sin\theta
\end{cases}$$
($\theta$ is the parameter, $m \neq 0$). If circle $C_1$ is externally tangent to circle $C_2$, then the value of the real number $m$ is ____.

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