
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
Suppose a parabola has a vertex at $\left(\frac{3}{5}, -\frac{13}{5}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $2a + b + 3c$ is an integer. Find the smallest possible value of $a.$

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