
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 \( a \), \( b \), and \( c \) be the distinct roots of the polynomial \( x^3 - 36x^2 + 215x - 470 \). Determine the value of \( \tfrac{1}{D}+\tfrac{1}{E}+\tfrac{1}{F} \) given that there exist real numbers \( D \), \( E \), and \( F \) such that 
\[
\dfrac{1}{t^3 - 36t^2 + 215t - 470} = \dfrac{D}{t-a} + \dfrac{E}{t-b} + \frac{F}{t-c}
\]
for all \( t\not\in\{a,b,c\} \).
A) 103
B) 104
C) 105
D) 106
E) 107

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