
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 a circle $C: x^2 + y^2 - 6x - 8y + m = 0$, where $m \in \mathbb{R}$.

(I) If circle $C$ is externally tangent to the circle $x^2 + y^2 = 1$, find the value of $m$;

(II) If the line $x + y - 3 = 0$ intersects circle $C$ and the length of the chord is $2\sqrt{7}$, find the value of $m$.

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