
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 that $f(x)$ is an even function defined on $R$, and $f(x)={\log_3}({{3^x}+1})-kx$, $g(x)=f(x)+\frac{1}{2}x$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Let $h(x)=x^{2}-2tx+5$. If there exists $x_{1}\in [\log _{3}2,8]$, for any $x_{2}\in \left[1,4\right]$, such that $g(x_{1})\leqslant h(x_{2})$, find the range of real number $t$.

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