
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 $AB$ be the diameter of a circle with center $O$ and $|AB| = 2R$. Let $C$ be a point on the circle and $M$ be a point on $AC$. A perpendicular $MN$ is dropped from $M$ to $AB$ and another perpendicular is erected to $AC$ intersecting the circle at point $L$ (the segment $CL$ intersects $AB$). Find the distance between the midpoint of $AO$ and the midpoint of $CL$, given that $|AN| = a$.

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