
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 $f(x) = a^{x-1}$ ($a>0$ and $a \neq 1$),  
(1) If the graph of the function $y=f(x)$ passes through point $P(3,4)$, find the value of $a$;  
(2) Compare $f(\lg \frac{1}{100})$ and $f(-2.1)$, and describe the comparison process;  
(3) If $f(\lg a) = 100$, find the value of $a$.

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