
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
If  $\phi$  is the Golden Ratio, we know that  $\frac1\phi = \phi - 1$ .  Define a new positive real number, called  $\phi_d$ , where  $\frac1{\phi_d} = \phi_d - d$  (so  $\phi = \phi_1$ ).  Given that  $\phi_{2009} = \frac{a + \sqrt{b}}{c}$ ,  $a, b, c$  positive integers, and the greatest common divisor of  $a$  and  $c$  is 1, find  $a + b + c$ .

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