
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
The Ferris wheel is a large amusement facility loved by many people. Visitors sit in the cabins of the Ferris wheel and slowly rotate upwards, enjoying the view from above. The radius of a certain Ferris wheel is $50$ meters, and the height of the center $O$ from the ground is $50$ meters. The Ferris wheel moves at a constant speed, completing one revolution every $3$ minutes. Point $P$ starts at the lowest point of the Ferris wheel. It is known that the height $y$ above the ground of a point $P$ on the Ferris wheel at time $t$ (in minutes) satisfies $y=A\sin \left(\omega t+\varphi \right)+b\left(\omega\ \ \gt 0,\varphi \in \left[-\pi ,\pi \right]\right)$<br/>$(1)$ Write the analytical expression of $y$ (in meters) with respect to $t$ (in minutes) based on the given conditions.<br/>$(2)$ If the best visual effect is achieved when the height above the ground reaches $85$ meters, how long does it take for the Ferris wheel to rotate for visitors to have the best visual effect?

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