
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 $tan(\alpha+\beta)=\frac{9}{13}$, $tan(\beta-\frac{\pi}{4})=-\frac{1}{3}$, $cos\gamma=\frac{{3\sqrt{10}}}{10}$, where $\alpha$ and $\gamma$ are acute angles.  
$(Ⅰ)$ Find the value of $\tan \alpha$;  
$(Ⅱ)$ Find the value of $\alpha +2\gamma$.

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