
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 $y=\log _{a}(2x+3)-4(a \gt 0$ and $a\neq 1)$ passes through a fixed point $P$, and the fixed point $P$ lies on the line $l:ax+by+7=0\left(b \gt 0\right)$. Find the minimum value of $\frac{1}{a+2}+\frac{1}{4b}$.

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