
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 the function $f(x)=kx+\log_3({3^x+1})(k\in R)$ is an even function.  
$(1)$ Find the value of the real number $k$.  
$(2)$ Determine the number of zeros of the function $g(x)=f(x)-\frac{1}{2}x-\log_3({3^x}-1)$.

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