
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 $f(x)=\log_{2}[(4^{x}+1)\cdot2^{kx}]$, where $k\in\mathbb{R}$ is an even function.  
$(1)$ Find the value of $k$;  
$(2)$ If $f(2t^{2}+1) < f(t^{2}-2t+1)$, find the range of $t$.

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