
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
Observe the following equations:

$$1-\frac{1}{2}=\frac{1}{2}$$,

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}=\frac{1}{3}+\frac{1}{4}$$,

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}=\frac{1}{4}+\frac{1}{5}+\frac{1}{6}$$,

$$\ldots$$

Based on this pattern, the $n$-th equation is $\_\_\_\_\_\_\_\_\_.$

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