
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 $f\left(x\right)=kx-\log _{4}(4^{x}+1)\left(k\in R\right)$ with its graph symmetric about the $y$-axis.<br/>$(1)$ Find the value of $k$;<br/>$(2)$ If the function $g\left(x\right)={4}^{\frac{1}{2}x-f(x)}-m\cdot 4^{1+kx}-1$,$x\in [0$,$\log _{2}3]$, find the maximum value of $g\left(x\right)$ denoted as $g\left(m\right)$.

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