
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 that a parabola has vertex  $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$ , and equation  $y=ax^2+bx+c$ , where  $a>0$  and  $a+b+c$  is an integer. The minimum possible value of  $a$  can be written as  $\tfrac{p}{q},$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .

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