
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 city plans to renovate its urban road network by adding $x$ new sections (Note: a section refers to a certain length of motor vehicle lane) and $n$ new road intersections on top of the existing $a$ sections, where $n$ and $x$ satisfy $n=ax+5$. It is known that the cost of building a new section is $m$ million yuan, and the cost of building a new road intersection is $k$ times the cost of building a new section.

$(1)$ Write the total cost $y$ (in million yuan) of building new road intersections as a function of $x$;

$(2)$ Let $P$ be the ratio of the total cost of building new sections to the total cost of building new road intersections. If the number of new sections is $20\%$ of the original number of sections, and $k \geqslant 3$, can $P$ be greater than $\frac{1}{20}$? Please explain your reasoning.

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