
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
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Vector $\overrightarrow{m}=(2a+c,b)$, $\overrightarrow{n}=(\cos B,\cos C)$, and $\overrightarrow{m}$, $\overrightarrow{n}$ are perpendicular.
(I) Determine the size of angle $B$;
(II) If the bisector of $\angle ABC$ intersects $AC$ at point $D$, and $BD=1$, set $BC=x$, $BA=y$, try to determine the function of $y$ with respect to $x$, and find the range of the length of side $AC$.

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