
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 $f(x)=a^{x}+\left(1-m\right)a^{-x}$ $(a \gt 0, a\neq 1)$ is an odd function and passes through the point $(-1,-\frac{3}{2})$.  
$(1)$ Find the values of real numbers $m$ and $a$.  
$(2)$ Let $g(x)=\log_{t}[2^{2x}+2^{-2x}-tf(x)]$ $(t \gt 0, t \neq 1)$. Determine if there exists a positive real number $t$ such that the inequality $g(x)\leqslant 0$ holds for all $x\in [1, \log_{2}3]$. If it exists, find the value of $t$; if not, explain the reason.

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