
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
Find the value of the expression $1+ \frac{1}{2+ \frac{1}{1+ \frac{1}{2+...}}}$ by employing an analogous method to the one used for calculating $\sqrt{1+ \sqrt{1+ \sqrt{1+...}}}$. In the latter, the following approach was utilized: let $\sqrt{1+ \sqrt{1+ \sqrt{1+...}}}=x$, hence $\sqrt{1+x}=x$, which leads to $x=\frac{1+\sqrt{5}}{2}$ (negative value is discarded).

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