
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 a point $A(0, a)$ on the $y$-axis where $a > 1$, for a moving point $M(x, y)$ on the curve $y = \left|\frac{1}{2} x^{2} - 1\right|$, find the minimum value of the distance $|AM|$ between the points $A$ and $M$ in terms of $a$.

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