
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
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\left(a-c\right)\left(\sin A+\sin C\right)=\left(a-b\right)\sin B$.  
$(1)$ Find the measure of angle $C$.  
$(2)$ If $D$ is the point that trisects side $AB$ (closer to point $A$) such that $CD=tAD$, find the range of real number $t$.

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