
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 car is driving towards a destination $180km$ away from the starting point. After the first hour of departure, it travels at a constant speed according to the original plan. One hour later, it travels at $1.5$ times the original speed and arrives at the destination $40$ minutes earlier than planned. Let the speed of the first hour be $x km/h$.  
$(1)$ Express the time it takes to complete the remaining distance after speeding up directly in terms of $x$ as ______$h$;  
$(2)$ Find the actual time it takes for the car to complete the entire journey;  
$(3)$ If the car returns along the original route, the driver plans to drive half of the distance at a speed of $a km/h$ and the other half at a speed of $b km/h$ $(a\neq b)$. Let the time taken be $t_{1}$ hours. If the driver drives at a speed of $a km/h$ for half the time and $b km/h$ for the other half, the time taken is $t_{2}$ hours. Compare the sizes of $t_{1}$ and $t_{2}$, and explain the reason.

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