
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
Omega and Theta both took part in a three-day problem-solving competition. At the end of the third day, each had attempted questions worth a total of 600 points. Omega scored 200 points out of 400 points attempted on the first day, and 150 points out of 200 points attempted over the next two days. Theta, who did not attempt the same number of points as Omega each day, had a positive integer score on each of the three days, and Theta's daily success ratio for each day was less than Omega's for that day. Omega's three-day success ratio was $\frac{200+150}{400+200} = \frac{350}{600} = \frac{7}{12}$.

Find the largest possible three-day success ratio that Theta could have achieved.

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