
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 function $f\left(x\right)=ax^{2}+x+c\left(a \gt 0\right)$ satisfies:<br/>① The function $f({x-\frac{1}{4}})$ is an even function;<br/>② The solution set of the inequality $f\left(x\right) \lt 0$ is $\left(m,1\right)\left(m \lt 1\right)$.<br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$;<br/>$(2)$ Find the minimum value $h\left(k\right)$ of the function $g\left(x\right)=f\left(x\right)+\left(4k+3\right)x\left(k\in R\right)$ on $\left[1,3\right]$.

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