
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 the diagram, a rectangle has a width $w$, a length of $10$, and a perimeter of $30$. What is the ratio of its width to its length? [asy]
pair a = (0, 0); pair b = (10, 0); pair c = (10, 5); pair d = (0, 5);
draw(a--b--c--d--cycle);
label("$w$", midpoint(a--d), W); label("$10$", midpoint(c--d), N);
[/asy] Write your answer in the form $x:y$, where $x$ and $y$ are relatively prime positive integers.

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