
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
Given vectors $\overrightarrow{m}=(a+c,b)$, $\overrightarrow{n}=(a-c,b-a)$, and $\overrightarrow{m}\perp \overrightarrow{n}$, where $A$, $B$, $C$ are the interior angles of $\triangle ABC$, and $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively.  
$(1)$ Find the size of angle $C$;  
$(2)$ Find the range of values for $\sin A+\sin B$.

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