
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 the function $f(x)=k\cdot 3^{x}+3^{-x}$ defined on $\mathbb{R}$ is an odd function.  
$(1)$ Find the value of the real number $k$;  
$(2)$ If for any $x\in \mathbb{R}$, the inequality $f[tx^{2}+(t-1)x]+f(x+t-3) \gt 0$ always holds, find the range of the real number $t$.

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