
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  $S$  be the set of all pairs  $(a,b)$  of real numbers satisfying  $1+a+a^2+a^3 = b^2(1+3a)$  and  $1+2a+3a^2 = b^2 - \frac{5}{b}$ . Find  $A+B+C$ , where \[ 
A = \prod_{(a,b) \in S} a
, \quad
B = \prod_{(a,b) \in S} b
, \quad \text{and} \quad
C =  \sum_{(a,b) \in S} ab.
\]*Proposed by Evan Chen*

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