
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
A chord intercepted by a circle on a line $l$ that passes through the point $(1,0)$ and is parallel to the line $x - \sqrt{2}y + 3 = 0$ has a length of _____. The equation of the circle is $(x - 6)^2 + (y - \sqrt{2})^2 = 12$.

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