
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
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $b=5$, $c=\sqrt{10}$, and $acosB+bcosA=\sqrt{2}ccosB$. Find:
- $(Ⅰ)$ the value of $a$;
- $(Ⅱ)$ given point $M$ lies on segment $BC$, if $tan∠AMB=\frac{3}{4}$, find the value of $\tan \angle MAC$.

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