
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
A certain local government aims to encourage entrepreneurship among the public by providing incentives to new small and micro enterprises with local output values ranging from $50,000$ to $500,000. The incentive scheme follows the following principles: the bonus $y$ (in units of $10,000) increases with the increase in annual output $x$ (in units of $10,000), and the bonus is not less than $70,000, while not exceeding $15\%$ of the annual output.  
$(1)$ If a certain enterprise has an output value of $100,000 and is entitled to a bonus of $90,000, analyze whether the function $y = \lg x + kx + 5$ (where $k$ is a constant) is a suitable model for the incentive function, and explain the reason (given $\lg 2 \approx 0.3, \lg 5 \approx 0.7$).  
$(2)$ If the function $f(x) = \frac{15x - a}{x + 8}$ is used as the incentive function model, determine the range of positive real numbers for $a$.

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