
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
Calculate the sum \(2^{x} + 2^{-x}\), given that \(4^{x} + 4^{-x} = 23\).

Simplify the following (7.16-7.17):

\[ 7.16 - \log_{2} \log_{2} \sqrt{\sqrt[4]{2}}. \]

Simplify, indicating the permissible values of the variables (7.18-7.19):

\[ 7.18 \left(2^{\log_{4} \sqrt{2}^{a}} - 3^{\log_{27} \left(a^{2} + 1\right)^{3}} - 2a\right) : \left(7^{4 \log_{49} a} - 5^{\frac{1}{\log_{5}^{5}} \sqrt{a}} - 1\right). \]

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