
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 $f(x)=a^{x}+a^{-x}$ $(a \gt 1)$, and $f(1)=\frac{10}{3}$.  
$(1)$ Find the value of the real number $a$;  
$(2)$ If the graph of $y=\log_{a}(\frac{m \cdot a^x - m}{f(x)})$ intersects the line $y=x$ at exactly one point, find the range of the real number $m$.

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