
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
(14 points) Given the functions $f(x) = x^2 + 1$ and $g(x) = 4x + 1$, both defined on the set $A$, the ranges of $f(x)$ and $g(x)$ are $S$ and $T$ respectively,  
① If $A = [1, 2]$, find $S \cap T$  
② If $A = [0, m]$ and $S = T$, find the value of the real number $m$  
③ If for any value of $x$ in set $A$, $f(x) = g(x)$, find the set $A$.

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