
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 that point $C$ is a moving point on line segment $AB$ of length 1, point $D$ is such that $DB \perp AB$ and $BD = AC$. Connect $AD$, and draw $CE \perp AD$ with the foot of the perpendicular at $E$. Let $AC = x$ and $BE = y$. Find the function expression of $y$ in terms of $x$, and determine the minimum value of $y$.

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