
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 \\(\alpha\\) and \\(\beta \in \left(\frac{3\pi}{4}, \pi\right)\\), \\(\sin(\alpha + \beta) = -\frac{3}{5}\\), \\(\sin\left(\beta - \frac{\pi}{4}\right) = \frac{12}{13}\\), find \\(\cos\left(\alpha + \frac{\pi}{4}\right) = \) ______.

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