
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 that $z$ is a solution to the equation $x^{2}+2x+10=0$ with $\text{Im}z < 0$, and $\frac{a}{z}+\overline{z}=bi$ (where $a$, $b$ are real numbers, $i$ is the imaginary unit, and $\text{Im}z$ denotes the imaginary part of $z$), find:

(I) the modulus of the complex number $w=a+bi$;

(II) the range of real values of $k$ such that the inequality $x^{2}+kx-a\geqslant 0$ holds for all $x\in[0,5]$.

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