
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
The vertices of a quadrilateral are \( A(0, 0) \), \( B(0, -1) \), \( C(1, 0) \), \( D\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \). Let \( K \) be the intersection point of the lines \( AB \) and \( CD \). Calculate the distance \( AK \).

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