
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
Let the function $f(x)=a^{x}+ka^{-x}$ where $a>0$ and $a\neq 1$ be an odd function, and $f(1)=\frac{3}{2}$.  
$(1)$ Find the values of $a$ and $k$;  
$(2)∃x∈[\frac{1}{2},2]$, such that the inequality $f(2x^{2})+f(1-mx) \lt 0$ holds, find the range of $m$.

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