
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
$f\left(x\right)=\log _{3}(a+x)+\log _{3}(6-x)$.  
$(1)$ If the graph of the function $f\left(x\right)$ is moved downward by $m\left(m \gt 0\right)$ and passes through $\left(3,0\right)$ and $\left(5,0\right)$, find the values of the real numbers $a$ and $m$.  
$(2)$ If $a \gt -3$ and $a\neq 0$, find the solution to the inequality $f\left(x\right)\leqslant f\left(6-x\right)$.

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