
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 real numbers $x$ and $y$ satisfying the equation $(x-a+1)^2 + (y-1)^2 = 1$, when $0 \leq y \leq b$ ($b \in \mathbb{R}$), the equation determines an even function $y=f(x)$. Then, the maximum distance from the focus $F$ of the parabola $y = -\frac{1}{2}x^2$ to a point on the trajectory of point $(a,b)$ is __________.

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