
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 $f(x)=\log_{2}(x+1)$.  
$(1)$ Move all points on the graph of function $f(x)$ to the right by $1$ unit to get the graph of function $g(x)$. Write the expression for function $g(x)$.  
$(2)$ If the function $y=g^{2}(x)-mg(x^{2})+3$ has a minimum value of $2$ on the interval $[1,4]$, find the value of $m$.

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