
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
Consider a multiplicative magic square formatted as follows:

$\begin{array}{|c|c|c|}
\hline 100 & a & b \\ \hline
c & d & e \\ \hline
f & g & 1 \\ \hline
\end{array}$

Each row, column, and diagonal have the same product $P$. Find the sum of all possible values of $f$ given that all entries are positive integers.
A) 120
B) 217
C) 235
D) 306

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