
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
Three squares have areas in the ratio $\frac{345}{45} : \frac{345}{30} : \frac{345}{15}$. After simplifying these ratios and rationalizing the denominators, the side lengths of these squares can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Find the sum $a+b+c$ for each square and determine the square with the maximum sum.

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