
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 that \(\alpha-l-\beta\) is a dihedral angle of \(45^\circ\), point \(C\) is within this dihedral angle with distances of 2 and 6 to the half-planes \(\alpha\) and \(\beta\) respectively. Points \(A\) and \(B\) are variable points on the half-planes \(\alpha\) and \(\beta\). Find the minimum perimeter of \(\triangle A B C\).

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