
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
Solve for real  $x$  :   $2^{2x} \cdot  2^{3\{x\}} = 11 \cdot  2^{5\{x\}} + 5  \cdot 2^{2[x]}$ (For a real number  $x, [x]$  denotes the greatest integer less than or equal to x. For instance,  $[2.5] = 2$ ,  $[-3.1] = -4$ ,  $[\pi ] = 3$ . For a real number  $x, \{x\}$  is defined as  $x - [x]$ .)

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