
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
Given the quadratic function $y=f(x)$, when $x=2$, the function $f(x)$ reaches its minimum value of $-1$, and $f(1)+f(4)=3$.

1. Find the analytic expression of $f(x)$;
2. If $g(x)=f(x)-kx$ has no minimum value in the interval $(1,4)$, find the range of the real number $k$.

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