
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
Given the quadratic function \( y = ax^2 + bx + c \) (where \( b \) is an integer), no point on the graph lies below the x-axis. The graph intersects the y-axis at point \( C \) and has its vertex at point \( E \). The axis of symmetry is to the right of the line \( x = c - \frac{1}{12} \). Let \( t = \frac{a + 2b + 12c}{a} \).

1. Find the minimum value of \( t \).
2. When \( t \) achieves its minimum value and \( CE > \frac{5}{24} \), determine the explicit form of the quadratic function.

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