
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
The quadratic polynomial \(p(x) = ax^2 + bx + c, \, a > 0\), when divided by \((x - 1)\) leaves a remainder of 4, and when divided by \((x - 2)\) leaves a remainder of 15. Find the maximum possible value of the ordinate of the vertex of the parabola \(y = p(x)\) under these conditions. For what value of \(x\) is this maximum achieved?

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