
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. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\cos A=\frac{7}{8}$, $c-a=2$, and $b=3$, find the value of $a$.
2. It is known that $2$, $m$, $n$, $p$, and $9$ form an arithmetic sequence. Find the value of $p-m$.
3. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $C=\frac{\pi}{3}$ and $c=\sqrt{3}$, find the maximum value of $a+b$.
4. In $\triangle ABC$, given that $AC=2\sqrt{2}$ and $BC=\sqrt{2}$, find the range of values for $\cos A$.

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