
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
Solve the equation \(3 \cdot 4^{x-2} + 27 = a + a \cdot 4^{x-2}\) and determine for which values of the parameter \(a\) it has solutions and for which values it does not.

Simplify the expressions and specify for which values of the variables the transformations are possible:

\[7.52 \left(b^{\frac{\log_{10} a}{\log a}} \cdot a^{\frac{\log_{100} h}{\log h}}\right)^{2 \log_{a}(a+b)}\]

\[7.53 \left(\left(\log_{b}^{4} a + \log_{a}^{4} b + 2\right)^{\frac{1}{2}} + 2\right)^{\frac{1}{2}} - \log_{b} a - \log_{a} b\]

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