
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 $f(x) = e^{x-1} - \frac{4a-3}{6x}$, $g(x) = \frac{1}{3}ax^2 + \frac{1}{2}x - (a-1)$.  
(1) The tangent line of the curve $f(x)$ at $x=1$ is perpendicular to the line $x+2y-1=0$. Find the value of the real number $a$.  
(2) When $x \geq 1$, $f(x) \geq g(x)$ always holds. Find the range of the real number $a$.

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