
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
Suppose that a sequence  $t_{0}, t_{1}, t_{2}, ...$  is defined by a formula  $t_{n} = An^{2} +Bn +c$  for all integers  $n \geq 0$ . Here  $A, B$  and  $C$  are real constants with  $A \neq 0$ . Determine values of  $A, B$  and  $C$  which give the greatest possible number of successive terms of the Fibonacci sequence.*The Fibonacci sequence is defined by*  $F_{0} = 0, F_{1} = 1$  *and*  $F_{m} = F_{m-1} + F_{m-2}$  *for*  $m \geq 2$ .

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