
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
For integers \( a \) and \( T, \) \( T \neq 0, \) a parabola whose general equation is \( y = ax^2 + bx + c \) passes through the points \( A = (0,0), \) \( B = (3T,0), \) and \( C = (3T + 1,35). \) Let \( M \) be the sum of the coordinates of the vertex point. Determine the largest value of \( M \).

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