
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
① The minimum value of the function f(x) is 1; ② The graph of the function f(x) passes through the point (-2,2); ③ The ordinate of the point where the graph of the function f(x) intersects the y-axis is 2. Choose one of these three conditions to complete the following question and solve it. <br />Question: For the quadratic function f(x) = ax^2 + bx + c that satisfies f(x+1) - f(x) = 2x + 3, and ____ (fill in the number of the selected condition). <br />(1) Find the analytical expression of f(x); <br />(2) Let the function $g(x) = f(x) - x^2 - x - 2 + \frac{k}{x}$, if $g(x) < -x + \frac{1}{x} + 6$ holds true for [1,4], find the range of real numbers for k.

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