
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 $f(x)$ satisfies $f(2x-3)=4x^{2}+2x+1$.  
$(1)$ Find the explicit formula of $f(x)$.  
$(2)$ Let $g(x)=f(x+a)-7x$, $(a\in\mathbb{R})$, try to find the minimum value of $g(x)$ on the interval $[1,3]$.

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