
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 line $l$: $kx + y + 4 = 0$ ($k \in \mathbb{R}$) is a symmetric axis of the circle $C$: $x + y^2 + 4x - 4y + 6 = 0$, and a line $m$ with a slope of 1 passes through point $A(0, k)$, calculate the length of the chord formed by the intersection of line $m$ and circle $C$.

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