
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 phenomenon of tides caused by the gravitational forces of the sun and moon on seawater is called tides. Generally, the tide that occurs in the morning is called flood tide, and the one that occurs in the evening is called ebb tide. In normal circumstances, ships sail into the channel and approach the dock during high tide; after unloading, they return to the ocean during low tide. The table below shows the relationship between the time and water depth at a certain port during a season:

| Time   | $2:00$ | $5:00$ | $8:00$ | $11:00$ | $14:00$ | $17:00$ | $20:00$ |
|--------|--------|--------|--------|---------|---------|---------|---------|
| Water Depth (m) | $7.5$ | $5.0$ | $2.5$ | $5.0$ | $7.5$ | $5.0$ | $2.5$ |

Through long-term observations, the relationship between the water depth and time at this port can be approximated by the function $f(t) = A\sin(\omega t + \varphi) + b$.

$(1)$ Based on the data above, find the expression for the function $f(t) = A\sin(\omega t + \varphi) + b$.

$(2)$ A cargo ship has a draft depth (distance between the ship's bottom and the water surface) of $4.25$ meters. Safety regulations require a minimum clearance of $2$ meters (distance between the ship's bottom and the ocean floor). During a day $(0:00-24:00)$, at what times can the cargo ship safely enter and exit the port?

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