
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 direction of the opening of the parabola $y=\frac{1}{2}(x-1)^2-1$ is ______, the coordinates of the vertex are (______, ______), and the axis of symmetry is ______. When $x$ is ______, the function $y$ increases as $x$ increases; when $x$ is ______, the function $y$ decreases as $x$ increases; at $x=$______, the function achieves the smallest ______ value, and the smallest $y$ value is $______$.

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