
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 circle \( C: (x - 2)^{2}+(y-2)^{2}=2 \) and the fact that it passes through the origin to form two tangents \( OT_{1} \) and \( OT_{2} \) at \( T_{1} \) and \( T_{2} \). Another line is drawn from the origin intersecting the line \( T_{1} T_{2} \) at point \( K \) and intersecting the circle at points \( M \) and \( N \). Given that \( |OM| = 2 \), find the length of \( ON \).

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