
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  $b$  and  $c$  be fixed real numbers and let the ten points  $(j,y_j )$  for  $j=1,2,\ldots,10$  lie on the parabola  $y =x^2 +bx+c.$  For  $j=1,2,\ldots, 9$  let  $I_j$  be the intersection of the tangents to the given parabola at  $(j, y_j )$  and  $(j+1, y_{j+1}).$  Determine the poynomial function  $y=g(x)$  of least degree whose graph passes through all nine points  $I_j .$ 

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