
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
Let \( p_{i} \) be the \( i \)th prime. Define the function

\[ f(x) = \sum_{i=1}^{50} p_{i} x^{i-1} = 2 + 3x + \cdots + 229x^{49} . \]

If \( a \) is the unique positive real number such that \( f(a) = 100 \), estimate \( A = \lfloor 100000a \rfloor \). An estimate of \( E \) will earn \( \max(0, \lfloor 20 - |A - E|/250 \rfloor) \) points.

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