
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

Let \(AMOL\) be a quadrilateral with \(AM = 10\), \(MO = 11\), and \(OL = 12\). Given that the perpendicular bisectors of sides \(AM\) and \(OL\) intersect at the midpoint of segment \(AO\), find the length of side \(LA\).

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