
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 local government plans to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between ¥500,000 and ¥5,000,000. The reward scheme follows these principles: the reward amount $y$ (in ten thousand yuan) increases with the increase of the annual output value $x$ (in ten thousand yuan), and the reward amount is not less than ¥70,000, while also not exceeding 15% of the annual output.  
$(1)$ If a company's output value is ¥1,000,000, and it is determined to receive a ¥90,000 reward, analyze whether the reward function model $y=\lg x+kx+5$ ($k$ is a constant) meets the government's requirements, 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 reward function model, determine the minimum value of the positive integer $a$.

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