
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
$19.$ The Guanyinshan Tunnel is $1165 m$ long, with a speed limit of $80 km/h$ (i.e., $\frac{200}{9} m/s$). A convoy of $17$ cars, each $5 m$ long (the cars can travel at a maximum speed of $40 m/s$), passes through the same lane of the tunnel at a constant speed. For safety and traffic flow, when $0 < x \leqslant 10$, the distance between adjacent cars is $20 m$; when $10 < x \leqslant \frac{200}{9}$, the distance between adjacent cars is $(\frac{1}{4} x^2 - \frac{1}{2} x) m$. The time taken from when the front of the first car enters the tunnel to when the back of the last car leaves the tunnel is denoted as $y$ seconds.

$(1)$ Express $y$ as a function of $x$;
$(2)$ Find the minimum time $y$ for the convoy to pass through the tunnel.

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