
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 $f\left(x\right)=\log _{a}(x-1)+2(a \gt 0$ and $a\neq 1)$ with its graph passing through the point $\left(3,3\right)$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Solve the inequality with respect to $x$: $f(2^{x}-3) \lt f(21-2^{x+1})$.

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