
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
(1) A symbol in the form of
$$
\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{vmatrix}
$$
is called a second-order determinant, and it is defined as
$$
\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{vmatrix}
= a_{11}a_{22} - a_{12}a_{21}.
$$
Calculate the value of the second-order determinant
$$
\begin{vmatrix}
\cos \frac{\pi}{4} & 1 \\
1 & \cos \frac{\pi}{3}
\end{vmatrix}
$$

(2) Given that $\tan\left( \frac{\pi}{4} + a \right) = -\frac{1}{2}$, find the value of
$$
\frac{\sin 2a - 2\cos^2 a}{1 + \tan a}.
$$

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