
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]


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Given positive real numbers $a$ and $b$ satisfying $4a+b=3$, find the maximum value of $b-\frac{1}{a}$ as ________; and the minimum value of $\frac{1}{3a+1}+\frac{1}{a+b}$ as ________.

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