
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
Functions  $f$  and  $g$  are defined so that  $f(1) = 4$ ,  $g(1) = 9$ , and for each integer  $n \ge 1$ ,  $f(n+1) = 2f(n) + 3g(n) + 2n $  and  $g(n+1) = 2g(n) + 3 f(n) + 5$ . Find  $f(2005) - g(2005)$ .

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