
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
The value of  $x$  which satisfies  $$ 1 +\log_x \left( \lfloor x \rfloor \right) = 2 \log_x \left(\sqrt3\{x\}\right) $$  can be written in the form  $\frac{a+\sqrt{b}}{c}$  , where  $a$ ,  $b$ , and  $c$  are positive integers and  $b$  is not divisible by the square of any prime. Find  $a + b + c$ .

Note:  $\lfloor x \rfloor$   denotes the greatest integer less than or equal to  $x$  and  $\{x\}$  denotes the fractional part of  $x$ .

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