
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
In class, the teacher posed the following question: Given $3a \gt b \gt 0$, $M=\frac{a}{b}$, $N=\frac{a+1}{b+3}$, compare the sizes of $M$ and $N$. Xiaohua: The comparison of polynomials can be done using the "difference method." Teacher: Compare $x^{2}+1$ and $2x-1$. Xiaohua: Since $(x^{2}+1)-(2x-1)=x^{2}+1-2x+1=(x-1)^{2}+1 \gt 0$, therefore $x^{2}+1 \gt 2x-1$. Teacher: Can the comparison of fractions be done using the "difference method"? ... (1) Please use the "difference method" to solve the teacher's question. (2) Compare the sizes: $\frac{23}{68}$______$\frac{22}{65}$. (Fill in "$ \gt $", "$=$", or "$ \lt $").

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