
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 perimeter of a rectangle $ABCD$ is $30$ inches, and the width $w$ of the rectangle is at least $6$ inches. The least value of diagonal $\overline{AC}$, in inches, is:
A) $10$ 
B) $7.5\sqrt{2}$
C) $15$
D) $\sqrt{225}$
E) $\text{none of these}$

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