
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 set $A=\{x \mid a-1 < x < 2a+1\}$, and the function $f(x) = ax + b$ ($a \neq 0$), satisfying $f(2x + 1) = 4x + 1$.
(1) Find the function $f(x)$.
(2) If the set $B=\{x \mid 1 < f(x) < 3\}$, and $B \subseteq A$, determine the range of the real number $a$.

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