
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
The determinant of the third order \\( \begin{vmatrix} -5 & 6 & 7 \\ 4 & 2^{x} & 1 \\ 0 & 3 & 1\end{vmatrix} \\) has an element \\(-5\\) whose algebraic cofactor is denoted as \\(f(x)\\). Solve for \\(x\\) in the equation \\(f(x)=0\\).

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